diff --git a/assets/css/main.css b/assets/css/main.css
index 57a51e7..da65e47 100644
--- a/assets/css/main.css
+++ b/assets/css/main.css
@@ -558,7 +558,7 @@ pre .k, pre .kd, pre .kn, pre .kp, pre .kr, pre .kt, pre .javascript .function {
 .links-of-author { margin-top: 20px; }
 
 .links-of-author a { display: inline-block; vertical-align: middle; margin-right: 10px; margin-bottom: 10px; border-bottom-color: #555; font-size: 13px; }
-.links-of-author a:before { display: inline-block; vertical-align: middle; margin-right: 3px; content: " "; width: 4px; height: 4px; border-radius: 50%; background: magenta; }
+.links-of-author a:before { display: inline-block; vertical-align: middle; margin-right: 3px; content: " "; width: 4px; height: 4px; border-radius: 50%; background: #47ffff; }
 
 .links-of-blogroll { font-size: 13px; }
 
diff --git a/atom.xml b/atom.xml
index d2367d2..394877c 100644
--- a/atom.xml
+++ b/atom.xml
@@ -1,4 +1,4 @@
-<?xml version="1.0" encoding="utf-8"?><feed xmlns="http://www.w3.org/2005/Atom" ><generator uri="https://jekyllrb.com/" version="3.9.0">Jekyll</generator><link href="routhleck.github.io/BrainPy-course-notes/atom.xml" rel="self" type="application/atom+xml" /><link href="routhleck.github.io/BrainPy-course-notes/" rel="alternate" type="text/html" /><updated>2023-08-28T05:19:28+00:00</updated><id>routhleck.github.io/BrainPy-course-notes/atom.xml</id><title type="html">BrainPy course notes</title><subtitle>Notes from the first training course on Neural Computational Modeling</subtitle><author><name>Sichao He</name></author><entry><title type="html">BrainPy course notes</title><link href="routhleck.github.io/BrainPy-course-notes/2023/08/27/BrainPy-course-notes/" rel="alternate" type="text/html" title="BrainPy course notes" /><published>2023-08-27T00:00:00+00:00</published><updated>2023-08-27T00:00:00+00:00</updated><id>routhleck.github.io/BrainPy-course-notes/2023/08/27/BrainPy-course-notes</id><content type="html" xml:base="routhleck.github.io/BrainPy-course-notes/2023/08/27/BrainPy-course-notes/">&lt;script src=&quot;https://polyfill.io/v3/polyfill.min.js?features=es6&quot;&gt;&lt;/script&gt;
+<?xml version="1.0" encoding="utf-8"?><feed xmlns="http://www.w3.org/2005/Atom" ><generator uri="https://jekyllrb.com/" version="3.9.0">Jekyll</generator><link href="routhleck.github.io/BrainPy-course-notes/atom.xml" rel="self" type="application/atom+xml" /><link href="routhleck.github.io/BrainPy-course-notes/" rel="alternate" type="text/html" /><updated>2023-08-28T06:56:50+00:00</updated><id>routhleck.github.io/BrainPy-course-notes/atom.xml</id><title type="html">BrainPy course notes</title><subtitle>Notes from the first training course on Neural Computational Modeling</subtitle><author><name>Sichao He</name></author><entry><title type="html">BrainPy course notes</title><link href="routhleck.github.io/BrainPy-course-notes/2023/08/27/BrainPy-course-notes/" rel="alternate" type="text/html" title="BrainPy course notes" /><published>2023-08-27T00:00:00+00:00</published><updated>2023-08-27T00:00:00+00:00</updated><id>routhleck.github.io/BrainPy-course-notes/2023/08/27/BrainPy-course-notes</id><content type="html" xml:base="routhleck.github.io/BrainPy-course-notes/2023/08/27/BrainPy-course-notes/">&lt;script src=&quot;https://polyfill.io/v3/polyfill.min.js?features=es6&quot;&gt;&lt;/script&gt;
 
 &lt;script id=&quot;MathJax-script&quot; async=&quot;&quot; src=&quot;https://cdn.jsdelivr.net/npm/mathjax@3/es5/tex-mml-chtml.js&quot;&gt;&lt;/script&gt;
 
diff --git a/image_url_process.py b/markdown_process.py
similarity index 57%
rename from image_url_process.py
rename to markdown_process.py
index 64abae6..50b9206 100644
--- a/image_url_process.py
+++ b/markdown_process.py
@@ -2,11 +2,7 @@
 import os
 
 
-def replace_img_url_1(input_path, out_path):
-    # 读取文件内容
-    with open(input_path, "r", encoding="utf-8") as file:
-        content = file.read()
-
+def replace_img_url_1(content):
     # 对于 ![image name](Notes.assets/image-000.png) 格式的替换
     pattern1 = r'!\[.*?\]\((Notes\.assets/.*?\.(png|jpg|jpeg|gif|bmp))\)'
     replacement1 = r'!\[.*?\](/assets/images/\1)'
@@ -17,16 +13,10 @@ def replace_img_url_1(input_path, out_path):
     replacement2 = r'<img src="/assets/images/\1"'
     content = re.sub(pattern2, replacement2, content)
 
-    # 将修改后的内容写入新文件
-    with open(out_path, "w", encoding="utf-8") as file:
-        file.write(content)
-
+    return content
 
-def replace_img_url_2(input_path, out_path):
-    # 读取文件内容
-    with open(input_path, "r", encoding="utf-8") as file:
-        content = file.read()
 
+def replace_img_url_2(content):
     # 对于 ![image name](Notes.assets/image-000.png) 格式的替换
     pattern1 = r'(!\[.*?\])\((/assets/images/Notes\.assets/.*?\.(png|jpg|jpeg|gif|bmp))\)'
     replacement1 = r'\1(/BrainPy-course-notes\2)'
@@ -37,14 +27,36 @@ def replace_img_url_2(input_path, out_path):
     replacement2 = r'<img src="/BrainPy-course-notes\1"'
     content = re.sub(pattern2, replacement2, content)
 
-    # 将修改后的内容写入新文件
-    with open(out_path, "w", encoding="utf-8") as file:
-        file.write(content)
+    return content
+
+
+def add_newlines_around_formula(content):
+    # 在 $$ 前后添加空行，确保不重复添加空行，并且公式块内部的内容不发生改变
+    pattern_start = r'(?<!\n)\n\$\$'
+    replacement_start = r'\n\n$$'
+    content = re.sub(pattern_start, replacement_start, content)
+    
+    pattern_end = r'\$\$(?!\n)'
+    replacement_end = r'$$\n'
+    content = re.sub(pattern_end, replacement_end, content)
+
+    return content
 
 
 if __name__ == "__main__":
     # 打印当前工作目录
     # print(os.getcwd())
-    input = "_posts/2023-08-27-BrainPy-course-notes.md"
-    out = "_posts/2023-08-27-BrainPy-course-notes_replace.md"
-    replace_img_url_2(input, out)
+    input_path = "_posts/2023-08-27-BrainPy-course-notes.md"
+    out_path = "_posts/2023-08-27-BrainPy-course-notes_replace.md"
+    
+    with open(input_path, "r", encoding="utf-8") as file:
+        md_content = file.read()
+    
+    md_content = replace_img_url_1(content)
+    md_content = replace_img_url_2(content)
+    md_content = add_newlines_around_formula(content)
+
+    # 将修改后的内容写入新文件
+        with open(out_path, "w", encoding="utf-8") as file:
+            file.write(md_content)
+