minor updates

entropy.tex

@@ -355,6 +355,16 @@ Shannon’s theorem is not constructive as it does not provide an explicit metho
 but it guarantees their existence.
 In practice, structured codes such as Hamming and Reed–Solomon codes are employed to approach channel capacity.
 
+\section{Conclusion}
+\section{Conclusion}
+Entropy provides a fundamental measure of uncertainty and information,
+bridging concepts from thermodynamics to modern communication theory.
+
+Beyond the provided examples, the concept of entropy has far-reaching applications in diverse fields:
+from cryptography, where it quantifies randomness and security, 
+to statistical physics, where it characterizes disorder in complex systems,
+to biology, connecting molecular information and population diversity.
+
 \printbibliography
 
 \end{document}