Fixed images, latex and first real blog.

.gitignore

@@ -1,3 +1,4 @@
 /public/
 /.packages/
 /.org-cache/
+/content/ltximg/

assets/css/site.css

@@ -106,7 +106,7 @@ body {
   font-size: 1.3rem;
   line-height: 1.5;
   color: #eeffff;
-  background-color: #000000;
+  background-color: #857470;
 }
 
 p {
@@ -154,7 +154,7 @@ h1 {
 }
 
 h2 {
-  color: #c11146;
+  color: #fc9600;
 }
 
 h3 {
@@ -339,7 +339,7 @@ strong {
   margin-top: 1rem;
   color: #93309f;
   font-size: 1rem;
-  background-color: #38033a;
+  background-color: #000000;
   border-top: 0.05rem solid #c3e88d;
 }
 

content/blogs.org

@@ -15,7 +15,8 @@
 
 * A list of All Blogs
 
-|-----------+------------------+------------------|
+|--------------------------------------------------+------------------+------------------|
-| Title     | Last Modified    | Created          |
+| Title                                            | Last Modified    | Created          |
-|-----------+------------------+------------------|
+|--------------------------------------------------+------------------+------------------|
-| [[/test_blog][Test Blog]] | [2024-08-17 Sat] | [2024-08-17 Sat] |
+| [[/countable_union_of_countable_sets_is_countable][A Countable Union of Countable Sets is Countable]] | [2024-12-06 Fri] | [2024-12-05 Thu] |
+| [[/test_blog][Test Blog]]                                        | [2024-08-17 Sat] | [2024-08-17 Sat] |

content/countable_union_of_countable_sets_is_countable.org

@@ -0,0 +1,178 @@
+#+title: A Countable Union of Countable Sets is Countable
+#+created: [2024-12-05 Thu]  
+#+last_modified: [2024-12-06 Fri]  
+#+author: Dibyashanu Pati
+#+OPTIONS: tex:dvipng
+#+OPTIONS: \n:t
+#+OPTIONS: toc:2
+
+* Union of Finite Countable Sets
+It is straight forward to show that the union of two countable set is countable.
+Let $S_0$ and $S_1$ be two countable infinite sets(in case when either is finite the proof trivial), that is, 
+
+\[\exists  f_0 : S_0 \rightarrow \mathbb{N}\] and
+
+\[\exists  f_1 : S_1 \rightarrow \mathbb{N}\]
+
+such that $f_0$ and $f_1$ are bijections,
+
+then the set $S = S_0 \cup S_1$ is also countable meaning that
+
+\[\exists f: S \rightarrow \mathbb{N}\] so that $f$ is a bijection.
+
+To construct such a map $f$ all the elements in $S_0$ to all the all the even naturals and all the elements in $S_1$ to the even naturals, so 
+
+\[S_0 = \{a_0, a_1 , a_2 , \hdots\}\]
+
+\[S_1 = \{b_0, b_1 , b_2 , \hdots\}\]
+
+let \[f(a_n) = 2n\] and \[f(b_n) = 2n + 1\]
+
+Such a map is surjective because we cover all cases modulo $2$ and hence all the integers, and this map is injective because $f$ is injective on all $a$'s and $b$'s separately.
+
+Either using induction or using a similar argument with the naturals modulo $k$ for any finite $k$ we can show that the union of any $k$ countable sets is also countable 
+
+$S = S_0 \cup S_1 \cup S_2 \hdots \cup S_k$.
+* Larger Unions?
+We know that an arbitrary union of countable sets is not necessarily countable. 
+Consider as a counterexample the union of all singletons $\{r\}, r \in \mathbb{R}$, this is $\mathbb{R}$ itself which we know not to be countable.   
+
+It turns out that a countable union of countable sets is countable, to show this we *cannot* use a induction or a modulo $k$ argument.
+
+The key idea in the modulo k argument is that there are $k$ equivalence classes and hence we can break the Naturals into $k$ infinite countable sequences(countable infinite sets), but can we break the naturals into infinitely many infinite sub-sequences? - Yes.
+
+The first time I encountered this it was supposed to be justified was by observing that the Naturals can be arranged into the following two dimensional array,
+
+\begin{matrix}
+\, \\
+0 & 1 & 3 & 6 & \dots \\
+2 & 4 & 7 & \dots \\
+5 & 8 & \dots \\
+9 & \dots \\
+\vdots
+\end{matrix}
+
+Although this is convincing, a thought process that would lead me to come up with this eluded me until now, I explain this thought process below.
+
+* The Argument 
+In the modulo $k$ argument we have a constant gap between each consecutive element of a sequence, because of this we are limited by the gap size($k$). So the key idea is constructing such a partitioning of the Naturals is to keep increasing the gap size, but since the gap size is finite at any point we can only have elements from a finite number of the countably infinite sub-sequences at any particular, so we are forced to start subsequent sub-sequences at larger and larger points.
+
+The most simple way to increase the gap size in this way is to keep increasing it by one after each gap. 
+
+[[pics/Countable_union_of_Countable_sets.jpg][image]]
+
+Look at the $0^{th}$ sequence we just get the [[https://en.wikipedia.org/wiki/Triangular_number][Triangular numbers]] and zero. 
+
+For elements of the first sequence we get the fill all places immediately after all Triangular numbers starting from the first Triangular number.
+
+For elements of the second sequence we get the fill all places one place after all Triangular numbers starting from the second triangular number.
+
+So we come up with the function $S$ that maps the $n^{th}$ element of the $m^{th}$ sub-sequence.
+
+$S(m,n) = m + \frac{(m+n)(m + n + 1)}{2}$
+
+This is equivalent to the 2D array I mentioned earlier.
+#+begin_src python :results output
+for m in range(0,5):
+    row = f" S_{m} = "
+    for n in range(0,5):
+        s = m + n
+        row += (str(m + int(s*(s+1)/2)) + ' , ')
+    print(row + ' ... \n')
+print('.\n.\n.\n')
+#+end_src
+
+#+RESULTS:
+#+begin_example
+ S_0 = 0 , 1 , 3 , 6 , 10 ,  ... 
+
+ S_1 = 2 , 4 , 7 , 11 , 16 ,  ... 
+
+ S_2 = 5 , 8 , 12 , 17 , 23 ,  ... 
+
+ S_3 = 9 , 13 , 18 , 24 , 31 ,  ... 
+
+ S_4 = 14 , 19 , 25 , 32 , 40 ,  ... 
+
+.
+.
+.
+
+#+end_example
+
+
+Now we just have to show that $S: \mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N}$  is bijective.
+
+------
+
+/Claim 1/: If $S(m_0, n_0) = S(m_1, n_1)$ then $m_0 + n_0 = m_1 + n_1$
+
+This equivalent to showing that if $m_0 + n_0 \ne m_1 + n_1$ then $S(m_0, n_0) \ne S(m_1, n_1)$.
+
+Without loss of generality let us assume  $m_0 + n_0 < m_1 + n_1$.
+
+Let $s_0 = m_0 + n_0$ and $s_1 = m_1 + n_1$ since $s_1>s_0$ the difference of there triangular numbers $T(s_1)$ and $T(s_0)$ is atleast $s_1$ because $s_0 \le s_1 - 1$.
+
+Then 
+\[
+S(m_1, n_1)  - S(m_0, n_0) &= \frac{s_1(s_1 + 1)}{2} - \frac{s_0(s_0 - 1)}{2} + m_1 - m_0
+\]
+\[
+ \ge s_1  + m_1 - m_0 
+\]
+\[ \ge s_1 - m_0
+\]
+\[ = m_1 + n_1 - m_0
+\]
+\[ > n_0
+\]
+\[ \ge 0
+\]
+
+So \[S(m_1, n_1)  > S(m_0, n_0)\]
+
+Similarly we can show that if $m_0 + n_0 > m_1 + n_1$ then $S(m_0, n_0)  > S(m_1, n_1)$ 
+
+------
+
+
+Now we show injectivity using /Claim 1/
+
+If $S(m_0, n_0) = S(m_1, n_1)$ then by /Claim 1/ $m_0 + n_0 = m_1 + n_1$
+
+Let $s_0 = m_0 + n_0$ and $s_1 = m_1 + n_1$  in this case Let $s = s_0 = s_1$
+
+So, 
+
+\[ 
+\implies S(m_1, n_1)  - S(m_0, n_0) &= \frac{s_1(s_1 + 1)}{2} - \frac{s_0(s_0 - 1)}{2} + m_1 - m_0 = 0 \]
+\[\implies 0 = 0 + m_1 - m_0 \]
+\[\implies m_1 = m_0 \]
+\[\implies s - m_1 = s - m_0 \]
+\[\implies n_1 = n_0 \
+\]
+
+Now we show surjectivity
+
+Let $N \in \mathbb{N}$
+
+let $T(s)$ be the largest triangular number less than $N$.
+
+Let $m = N - T(n)$
+
+Let $n = s - m$
+
+Then
+
+ $S(m,n) = N$ 
+
+Since $S: \mathbb{N} \times \mathbb{N} \rightarrow \mathbb{N}$ is both injective and surjective, it is a bijection, we are done.
+
+* An Interesting Applications 
+*The set of finite subsets of $\mathbb{N}$ is countable.* 
+Let $A_i$ be set of all subsets of $\mathbb{N}$ containing $i$ elements,  that is with cardinality $i$.
+
+$A_1$ is just the set of all singletons of $\mathbb{N}$, there is an obvious bijection of this with $\mathbb{N}$, the identity map - hence it is countable.
+Now we just use induction,
+Let $A_k$ be countable, then $A_{k + 1} = \bigcup\limits_{i \in \mathbb{N}} \{S \cup i : i \notin S \land S \in A_k \}$ is a countable union of countable sets.
+Since all all $A_i$'s are countable the set of all finite subsets $A = \bigcup A_i$ being a countable union of countables is also countable. 

content/index.org

@@ -6,5 +6,11 @@ Cosmicflow is a site devoted to the interests of Dibyashanu Pati.
 
 I'm currently a student, I love Physics, Mathematics, Astronomy and Computers.
 
+* Recent Blogs
+|--------------------------------------------------+------------------+------------------|
+| Title                                            | Last Modified    | Created          |
+|--------------------------------------------------+------------------+------------------|
+| [[/countable_union_of_countable_sets_is_countable][A Countable Union of Countable Sets is Countable]] | [2024-12-06 Fri] | [2024-12-05 Thu] |
+
 
 

create-site.el

@@ -198,18 +198,37 @@
                                (file-name-sans-extension
                                 (org-element-property :path link)))))
 
-  (let ((exported-link (org-export-custom-protocol-maybe link contents 'html info)))
+;;   (let ((exported-link (org-export-custom-protocol-maybe link contents 'html info)))
-    (cond
+;;     (cond
-     (exported-link exported-link)
+;;      (exported-link exported-link)
-     ((equal contents nil)
+;;      ((equal contents nil)
-      (format "<a href=\"%s\">%s</a>"
+;;       (format "<a href=\"%s\">%s</a>"
-              (org-element-property :raw-link link)
+;;               (org-element-property :raw-link link)
-              (org-element-property :raw-link link)))
+;;               (org-element-property :raw-link link)))
-     ((string-prefix-p "/" (org-element-property :raw-link link))
+;;      ((string-prefix-p "/" (org-element-property :raw-link link))
-      (format "<a href=\"%s\">%s</a>"
+;;       (format "<a href=\"%s\">%s</a>"
-              (org-element-property :raw-link link)
+;;               (org-element-property :raw-link link)
-              contents))
+;;               contents))
-     (t (org-export-with-backend 'html link contents info)))))
+;;      (t (org-export-with-backend 'html link contents info))))
+(let ((exported-link (org-export-custom-protocol-maybe link contents 'html info)))
+  (cond
+   (exported-link exported-link)
+   ((equal contents nil)
+    (format "<a href=\"%s\">%s</a>"
+            (org-element-property :raw-link link)
+            (org-element-property :raw-link link)))
+   ((string-prefix-p "/" (org-element-property :raw-link link))
+    (format "<a href=\"%s\">%s</a>"
+            (org-element-property :raw-link link)
+            contents))
+   ((and (string-match-p (concat "\\." (regexp-opt '("jpg" "jpeg" "png" "gif")))  
+         (org-element-property :raw-link link))
+         (not (equal contents nil)))
+    (format "<img src=\"%s\" alt=\"%s\" style=\"max-width: 100%%; height: auto;\">"
+        (org-element-property :raw-link link)
+        contents))
+   (t (org-export-with-backend 'html link contents info))))
+)
 
 
 (defun my/make-heading-anchor-name (headline-text)
@@ -256,25 +275,25 @@
 	       (code (org-html-format-code src-block info)))
     (format "<pre>%s</pre>" (string-trim code))))
 
-(defun my/org-html-special-block (special-block contents info)
+;; (defun my/org-html-special-block (special-block contents info)
-  "Transcode a SPECIAL-BLOCK element from Org to HTML.
+;;   "Transcode a SPECIAL-BLOCK element from Org to HTML.
-CONTENTS holds the contents of the block.  INFO is a plist
+;; CONTENTS holds the contents of the block.  INFO is a plist
-holding contextual information."
+;; holding contextual information."
-  (let* ((block-type (org-element-property :type special-block))
+;;   (let* ((block-type (org-element-property :type special-block))
-         (attributes (org-export-read-attribute :attr_html special-block)))
+;;          (attributes (org-export-read-attribute :attr_html special-block)))
-	  (format "<div class=\"%s center\">\n%s\n</div>"
+;; 	  (format "<div class=\"%s center\">\n%s\n</div>"
-            block-type
+;;             block-type
-            (or contents
+;;             (or contents
-                (if (string= block-type "cta")
+;;                 (if (string= block-type "cta")
-                    "If you find this guide helpful, please consider supporting System Crafters via the links on the <a href=\"/how-to-help/#support-my-work\">How to Help</a> page!"
+;;                     "If you find this guide helpful, please consider supporting System Crafters via the links on the <a href=\"/how-to-help/#support-my-work\">How to Help</a> page!"
-                  "")))))
+;;                   "")))))
 
 (org-export-define-derived-backend 'site-html 'html
   :translate-alist
   '((template . my/org-html-template)
     (link . my/org-html-link)
     (src-block . my/org-html-src-block)
-    (special-block . my/org-html-special-block)
+    ;; (special-block . my/org-html-special-block)
     (headline . my/org-html-headline))
   :options-alist
   '((:video "VIDEO" nil nil)))

create.sh

@@ -2,3 +2,6 @@
 
 # generate the html and gemini versions of the site using org-publish
 emacs -Q  --batch -l create-site.el  
+
+rsync -avi ./content/ltximg ./public/cosmicflow-html/
+# rm -rf ./content/ltximg/