A semi-submarine is a type of marine vessel that operates partially submerged underwater, combining features of both submarines and surface ships. These vessels typically have a hull design that allows them to float on the surface while also having the capability to dive to certain depths.

Prime numbers can be categorized into various classes based on their properties and characteristics. Here are some of the most commonly recognized classes of prime numbers: 1. **Regular Prime Numbers**: These are the standard prime numbers greater than 1 that are only divisible by 1 and themselves (e.g., 2, 3, 5, 7, 11, etc.). 2. **Even Prime Numbers**: The only even prime number is 2.

A Cunningham chain is a sequence of prime numbers such that each prime (after the first) can be formed by doubling the previous prime and then adding or subtracting one.

There is no simple formula that generates all prime numbers, nor is there a formula that can predict the nth prime number efficiently. However, there are several interesting approaches and formulas that either generate primes or are related to primes. Here are a few notable ones: 1. **Wilson's Theorem**: A prime number \( p \) satisfies the equation: \[ (p-1)!

PrimePages is a website dedicated to the study and exploration of prime numbers. It serves as a resource for enthusiasts, mathematicians, and anyone interested in the properties of prime numbers. The site typically features information about large prime numbers, including discoveries and records, as well as discussions on prime-related topics like primality testing, prime factorization, and the distribution of primes.

The prime-counting function, denoted as \( \pi(x) \), is a mathematical function that counts the number of prime numbers less than or equal to a given number \( x \).

"The Drunkard's Walk: How Randomness Rules Our Lives" is a book by Leonard Mlodinow, published in 2008. In this work, Mlodinow explores the concept of randomness and how it affects our everyday decisions and experiences. The title refers to the mathematical concept of a "random walk," a path that consists of a series of random steps, often used in probability theory and statistics.

"The Music of the Primes" is a book by mathematician Marcus du Sautoy, published in 2003. The book explores the enigmatic world of prime numbers and their significance in mathematics, particularly in number theory. Du Sautoy delves into the historical context of the study of prime numbers, discusses various mathematical theorems and concepts, and introduces readers to key figures who have contributed to this field.

"Modern Stochastics: Theory and Applications" is a book that generally focuses on the field of stochastic processes and their applications. The book typically covers probabilistic models and techniques that are used to analyze random phenomena in various disciplines, such as finance, insurance, telecommunications, and more. The content of this book may include topics like: 1. **Basic Probability Theory**: Foundational concepts that underpin stochastic processes.

The "Annals of Probability" is a peer-reviewed scientific journal that publishes research articles in the field of probability theory and its applications. Established in 1973, the journal focuses on a wide range of topics within probability, including stochastic processes, random walks, and statistical mechanics, among others. It serves as a platform for researchers to disseminate their findings and contribute to the development of probabilistic methods and theories.

Hans Christian von Baeyer is a physicist and author, known for his work in various areas of physics and for writing popular science books. He has contributed to the fields of quantum mechanics, optics, and condensed matter physics, among others. Additionally, von Baeyer has written extensively to communicate scientific concepts to the general public, making complex ideas accessible and engaging. One of his notable books is "Taming the Atom," which discusses the scientific and philosophical implications of atomic theory.

Alice Guionnet is a prominent mathematician known for her work in the fields of probability theory and statistics. She has made significant contributions to areas such as stochastic processes and statistical inference. Guionnet has published numerous research papers and is recognized for her contributions to understanding complex systems and their behaviors. She is also involved in academic activities, including teaching and mentoring students in mathematics and related disciplines.

Carl-Gustav Esseen was a Swedish mathematician, best known for his contributions to probability theory and statistics. He made significant advancements in the field, particularly related to the central limit theorem, which describes the distribution of the sum of a large number of independent random variables. One of his notable works is the development of an inequality that provides a quantitative measure of how closely a distribution approximates a normal distribution, known as the Esseen's bound or Esseen's inequality.

Elizabeth Meckes is a professor of mathematics at Case Western Reserve University, known for her work in areas such as computational mathematics, partial differential equations, and mathematical biology.

Carlo Emilio Bonferroni was an Italian mathematician, best known for his contributions to the field of statistics. He was born on January 18, 1892, and passed away on March 18, 1960. Bonferroni is most noted for the Bonferroni correction, a statistical adjustment used to address the problem of multiple comparisons. This correction helps to reduce the chances of obtaining false-positive results when multiple hypotheses are tested simultaneously.

Donald A. Dawson is a prominent figure in the field of mathematics, particularly known for his work in the areas of stochastic processes and applied probability. He has made contributions to topics such as stochastic calculus, particle systems, and interacting particle systems. Dawson is often cited in academic literature, and his work has influenced various applications in fields like finance, physics, and biology. If you were referring to something specific related to Donald A. Dawson, such as a particular paper or achievement, please provide more context.

Geoffrey Grimmett is a prominent British mathematician known for his work in the fields of probability theory and mathematical statistics, particularly in the study of percolation theory and random processes. He has made significant contributions to the understanding of random graphs and statistical mechanics. Grimmett has also authored several influential books and papers on these topics and is known for his teaching and research in mathematical sciences.

Frank Spitzer could refer to different individuals, depending on the context. One notable Frank Spitzer is Frank Spitzer (1928–2017), an American mathematician known for his work in probability theory and mathematical statistics. He is noted for his contributions to various areas, including the field of stochastic processes.

George Uhlenbeck (born December 6, 1900 – died October 31, 1988) was a notable Dutch-American physicist best known for his contributions to quantum mechanics. He is particularly recognized for his role in the development of the concept of spin in quantum theory, which describes an intrinsic form of angular momentum possessed by elementary particles.

Gesine Reinert is a notable figure in the field of computer science, particularly known for her work in bioinformatics and computational biology. As of my last knowledge update, she has been involved in developing algorithms and methods for analyzing biological data, particularly related to DNA, RNA, and protein sequences. Reinert has contributed to various research projects and has published numerous papers in scientific journals, focusing on applications of computational techniques to solve biological problems.