As a programming specialist with extensive experience, I've been immersed in the field of web development for a considerable time. Despite not having an academic background in IT or programming, I have acquired proficiency in several languages such as PHP and Python, and have successfully carried out projects of varying scales.



However, I have come to realize that mastering a programming language and staying updated through online resources are just the tip of the iceberg. There exists a significant gap in my fundamental knowledge, typically covered in university courses like mathematical analysis, number theory, and algorithms. I am eager to enhance my expertise by delving into these subjects. Could you guide me on the key areas and topics that I should focus on for self-improvement? Additionally, recommendations on relevant literature would be greatly appreciated.

I prefer not to pursue formal education, opting instead to learn independently and apply the knowledge directly. I believe this approach will expedite my learning process compared to attending lectures for an extended period. Despite searching extensively online, I have found the information to be either insufficient or unclear regarding the specific content covered in these academic disciplines and their relevance to programming.

I extend my gratitude in advance to all who provide assistance.

P.S. If you could offer multiple suggestions, please prioritize them in terms of their sequential importance for study. Alternatively, you could outline the foundational topics that need to be mastered first before progressing to others.

UPD: Please refrain from suggesting "reading textbooks on programming languages," as I have already covered this extensively at the outset of my journey. I am specifically interested in aspects of programming that transcend language-specific knowledge.

I would suggest focusing on the following key areas and topics:

1. Algorithms and Data Structures:
  Delve into the fundamental principles of algorithms, data structures, and their implementation. Understand concepts such as searching, sorting, and graph algorithms. Resources like "Introduction to Algorithms" by Thomas H. Cormen can provide a solid foundation.

2. Mathematical Analysis:
  To bolster your understanding of mathematical concepts relevant to programming, consider topics like calculus, discrete mathematics, and linear algebra. This will aid in grasping the theoretical underpinnings of many algorithms. "Concrete Mathematics" by Ronald L. Graham is a recommended text to explore.

3. Number Theory:
  Although not directly applicable in every programming scenario, number theory can strengthen your problem-solving skills and logic. Familiarize yourself with prime numbers, modular arithmetic, and cryptography. "An Introduction to the Theory of Numbers" by G.H. Hardy is a classic resource in this domain.

4. Software Engineering Principles:
  Gain insight into software design patterns, architecture, and best practices. Understanding concepts like SOLID principles and design patterns such as MVC can significantly impact the quality of your code. "Design Patterns: Elements of Reusable Object-Oriented Software" by Erich Gamma provides an excellent overview.

5. System Design:
  Explore the concepts of scalability, distributed systems, and system architecture. This knowledge will be invaluable when working on larger-scale projects. "Designing Data-Intensive Applications" by Martin Kleppmann is a comprehensive guide in this area.

By prioritizing these sequential topics, you can systematically build a strong foundation for self-improvement in programming beyond language-specific knowledge. Remember that practical application of these concepts is equally important, so seek opportunities to implement what you learn in real-world projects.

It's crucial to continually test and enhance our memory, as certain details require constant attention. With the vast amount of information available, a good memory combined with fast reading skills is invaluable. One resource for improvement is William Atkinson's book "The Science of Self-Improvement and Influencing Others," though it may be a bit outdated.

Next, the ability to concentrate and exercise willpower is essential. The brain's capacity is finite and needs proper utilization to maximize productivity. Distractions, such as music not intended for musicians, can significantly impact the quality of one's work. Therefore, mastering concentration and willpower is vital for maintaining focus and preventing dispersion of mental energy.

When it comes to handling information, conceptual understanding is key, rather than mere memorization. Organizing information is a crucial skill for any intellectual worker. To truly master your craft, imposing limitations on oneself can be beneficial. This could mean narrowing the scope of your work and embracing the role of an apprentice in related fields.

Lastly, it's important to consider the broader perspectives, such as "Athene's Theory of Everything," for insights into modern scientific thought about brain function. Though it may seem cliché, the aspects of memory, concentration, and information management are undeniably vital for any intellectual work specialist.

I believe that a web programmer should first and foremost be a skilled programmer. Therefore, I highly recommend N. Wirth's "Algorithms and Data Structures" as essential reading. This book provides fundamental knowledge and is extremely beneficial, even though it may be considered an ABC or second course text.

It's crucial to delve into tree structures and algorithms on graphs. When I was younger, Kubensky's book had a profound impact on my understanding of these concepts.

I also suggest researching genetic algorithms, you can find valuable information on Google.

Another valuable resource is "The Pearls of Programming" by J. Bentley. This book offers insightful analysis of programming problems and provides non-trivial solutions.

For the rest, begin with what you already know about your programming language, tools, libraries, and frameworks. Do you utilize associative arrays? Are you familiar with how they function, and what about key access? How time-consuming is this process?

Have you used TreeMap? The javadoc mentions red-black trees in relation to it. What does this mean, and how does it relate to your work?

Understanding how a program is executed in your chosen language is also vital. If you need to integrate a SQL-like reporting language into your application, how would you approach this?

If you can articulate the structure and principles of your programming tools and predict their behavior, evaluate the complexity of different approaches, then you are no longer just self-taught.

Discrete mathematics, particularly combinatorics, graph theory, and number theory, will provide a solid foundation for understanding algorithms and data structures. Next, delve into linear algebra, as it's essential for working with vectors and matrices in programming. Additionally, grasp the basics of calculus, including derivatives and integrals, which will help you understand optimization techniques and machine learning concepts.

To supplement your learning, I suggest exploring online resources like Coursera, edX, and MIT OpenCourseWare, which offer courses on these topics. For a more in-depth understanding, consider textbooks like "Introduction to Algorithms" by Thomas H. Cormen, "Discrete Mathematics with Applications" by Kenneth H. Rosen, and "Linear Algebra and Its Applications" by Gilbert Strang.