Introduction
possible combinations. This can be used for estimating the security of some protection methods. However, the applications of combinatorics span significantly further, overlapping with fields such as computer science, statistics, chemistry, and others.
Combinatorics in Computer Science
Computer science is a field where combinatorics has the highest number of applications. Most related problems in this sphere involve the use of graphs. Graph theory is an integral part of combinatorics that helps to visualize and understand complex situations. It represents certain objects as vertices connected by edges that represent their relationships. The vertices can have different numbers of connections, and the edges can be bidirectional or unidirectional. In computer science, the edges of graphs can have weights, which presents an extensive list of computational problems and widens the range of uses of Combinatorics.
Modeling tasks using graphs with weighted edges allows computer scientists to estimate the complexity of their algorithms by predicting the amount of time or computational resources required to execute them. A program can automatically analyze the graph to find a path with minimal total weight, showing the most efficient way of performing a given task. A graph can also be viewed as a network of pipes with defined restrictions that can be used to calculate the peak flow of a system.
Another part of combinatorics that is widely used in computer science is complexity theory. Complexity theory is a development of the systems theory that focuses on the uncertain and non-linear properties of systems that are partially controlled by pre-defined rules. In more practical terms, computational complexity theory is used to assess the difficulty of solving combinatorics-related problems. An example of a problem that falls into the domain of the complexity theory would be determining whether a given propositional formula has a satisfying assignment (Dean, 2016). Complexity theory tries to formalize problems such as the one mentioned above by assigning them to classes based on the approaches that can be used to solve them.
In addition to optimizing algorithms, combinatorial designs are used in designing computer hardware that would be used to perform these, and other tasks. The wide use of combinatorics has allowed systems engineers to achieve a more precise balance between the amount of resources required, the manufacturing complexity, and the effectiveness of the finished product. The role of this branch of mathematics in computer science continues to evolve, significantly simplifying many problems that would otherwise have been extremely challenging to solve.
Combinatorics in Chemistry
The combination of chemistry and combinatorics is commonly known as combinatorial chemistry. Similarly to computer science, the use of combinatorics in chemistry unlocks a broad range of previously unknown possibilities. Combinatorial chemistry’s main achievement is the possibility of synthesizing remarkably complex compounds in a single highly efficient process. The main principle behind combinatorial chemistry is the creation of vast libraries of possible compounds and then analyze them to find ones that might have a practical use. At the inception of this method, many researchers believed that it would replace traditional research as the primary source of discovery of new molecules. Currently, combinatorial chemistry is mainly used in medical applications, although naturally sourced molecules still account for a major portion of new pharmaceutical research.
Combinatorial chemistry offers unmatched possibilities in designing synthetic materials, however, it cannot replace natural compounds in some applications. Its main advantage is the fact that it gives modern chemists access to a diverse and unpredictable range of viable options. One might find it surprising that natural sources also boast an unprecedented amount of options that can be used in the pharmaceutical field. Furthermore, it is believed that science has only discovered a small portion of what natural sources have to offer.
The niche of combinatorial chemistry seems to be in simplifying the task of optimizing the process of designing complex molecules. This approach can automatically analyze all possible combinations within the required criteria and help select the optimal variant. In addition to the “brute force” approach to discovering new molecules, combinatorics is often used to quickly test all derivatives and permutations of a promising compound. This has enabled chemists to develop a wide spectrum of substances without producing excessive waste.
Apart from its vast range of applications in the pharmaceutical field, combinatorial chemistry is also used in agriculture. It plays a significant role in the creation of safer and more effective pesticides, insecticides, and substances that can help rid plants of harmful fungi. Moreover, combinatorial chemistry is responsible for the development of compounds that are used to control the growth of certain plants. Similar to the pharmaceutical sector, the agricultural industry will likely see numerous new discoveries thanks to combinatorics.
Combinatorics in Statistics
Combinatorial statistics is a field that involves working with large datasets. This area is concerned with questions such as the distribution of the statistics, or the average value in a given set. It analyzes given information and estimates uncertain numbers, allowing professionals in the field to make more accurate and grounded judgments. In some cases, combinatorics can be used to make predictions based on patterns in existing data, and known conditions that may affect it in the future. Combinatorics has a major effect on the study of statistics, as it is known to be the basis of the most crucial aspects of the latter. For instance, some of the most popular divisions of statistics that are rooted in combinatorics and mathematical problems include non-parametric statistics, statistical distribution theory, queuing theory, and the study of urn models.
Combinatorics in Physics
Combinatorial physics uses mathematics and combinatorics to find solutions to challenging physical problems requiring complex calculations. One example of a physical problem that requires combinatorics would be finding the way particles are distributed within a system to a statistical degree of accuracy. Combinatorial physics is able to use statistical distribution functions that are present in both classic and quantum physics to find the occupational probability of a given energy range. Combinatorics also plays a substantial role in theoretical physics. It enables researchers to make impressive progress in the quantization of strings and fields and is integral to any work related to the quantum field theory.
Combinatorial physics is a relatively new field, uniting mathematics and physics to achieve previously unreachable scientific milestones. Although its main scope currently lies in quantum theory, combinatorics might find increasing use in mechanics, as machines become more complex. The present level of technology does not require the use of combinatorial physics in most practical applications. However, as research in quantum theory progresses, there will be engineering challenges requiring the use of combinatorics.
Combinatorics in Finance
Stock trading is one of the most lucrative uses of combinatorics, as it directly results in increased profits. A high percentage of trading operations in the world’s leading exchanges are conducted by computers without human input. This phenomenon is known as high-speed trading, thanks to the incredibly quick reaction times of the highly attuned trading systems. With fully autonomous computer-controlled trading, it is crucial to have a reliable way to make the decision of what to buy and what to sell. The unpredictable and speculative nature of stock trading requires the systems to analyze all available data to estimate potential risks as accurately as possible, in order to consistently generate profits.
The first way in which combinatorics helps traders is by optimizing their systems. It can be used to analyze the code and determine how much time certain operations would take, opening opportunities for increasing efficiency. The second way that combinatorics can be implemented in stock trading using it to improve decision-making. Predictive models can employ combinatorial methods to determine whether a certain purchasing decision is likely to result in a profit or a loss. It is also possible to use combinatorics to rapidly generate viable alternative strategies in order to maintain a competitive edge.
Combinatorics in Biology
The final field involving combinatorics that will be reviewed in this paper is combinatorial biology. Similar to combinatorial chemistry, it primarily works with creating new compounds. It also uses biosynthetic methods for producing the compounds, however, this area of research focuses mainly on gene research, rather than the creation of medical or agricultural products. Combinatorial biology enjoys the same benefits of rapidly designing batches of compounds based on desired parameters, and the consequent increase in efficiency.
The technology that enables combinatorial biology to reach its full potential is known as high-throughput screening. This method involves the use of robots and other specialized equipment to automate biological and chemical testing. The rate at which a high-throughput screening system operates allows researchers to test millions of compounds within a reasonable timeframe. A researcher that is testing a newfound property of some gene can quickly test all related combinations and permutations of the originally discovered compound to find variants that better meet their criteria.
Conclusion
Combinatorics is used in a vast array of fields, from natural sciences such as physics, chemistry, and biology, to practically oriented disciplines such as computer science, statistics, and finance. It is a unique science in the way it offers methods that can significantly boost research in a wide spectrum of areas. The possibilities that have been opened by combinatorial approaches to existing problems have drastically improved many aspects of human life. Its impact on computer science laid the foundation for all other progress, as it resulted in the creation of more powerful software and hardware that is used in other fields. The latest research in chemistry, and specifically the pharmaceutical sector, has allowed people to live longer and healthier lives. Combinatorial biology has made numerous discoveries in gene modification, which have had a major positive effect on the agriculture industry. Finally, the use of combinatorics in finance is driving the economic development of the modern world.
The current achievements of combinatorics in other fields are remarkable, however, this sector likely has greater potential. The most promising application of combinatorics is in quantum physics, due to the fact that this field offers something that is principally different from every other tool available to humanity. Quantum computing is currently in the very initial stage of development and has little practical value. With time, it will be able to assist people with tasks that traditional computers cannot solve in any amount of time. It is also likely to open access to new research areas that are currently outside of most people’s comprehension. The inevitable leap in computational technology will cause breakthroughs in other fields, which would in turn yield global quality of life improvements. While combinatorics may not be at the forefront of all innovation, it is a powerful and versatile tool that serves as a driving force of progress in key areas. Its impact on the world we live in is not obvious, but it should not be underestimated.
References
Dean, W. (2016). Computational Complexity Theory. Stanford Encyclopedia of Philosophy. Web.