Gordon Royle is a notable mathematician, particularly recognized for his work in the field of combinatorics and graph theory. He is a professor at the University of Western Australia and has contributed to various areas including the study of matroids, combinatorial designs, and graph structures. Additionally, he is known for his involvement in the development of computer-based techniques for enumerating graphs and designs, and he has published numerous research papers in these areas.

As of my last update in October 2023, there isn't a prominent mathematician widely recognized by the name John H. Smith. It's possible that he may be a lesser-known figure in the field, or there may be multiple individuals with that name in various academic contexts. If you're looking for information about a specific John H. Smith who has contributed to mathematics, could you provide any additional context or details? This would help in identifying the right person or contributions.

Mark Ellingham is a British travel publisher and the founder of the renowned travel guide company, *Rough Guides*. Established in the 1980s, *Rough Guides* is known for its comprehensive and insightful travel publications, which cover various destinations around the world. Ellingham has been influential in shaping the travel publishing industry, promoting a more adventurous and independent approach to travel. The guides are recognized for their detailed information, practical tips, and emphasis on cultural immersion.

Narsingh Deo is a historical figure from India, typically associated with the region of Odisha. He is often recognized as a significant ruler of the Ganga dynasty during the medieval period. The Ganga dynasty was known for its contributions to art, architecture, and culture, particularly in the temple town of Puri, which is famous for the Jagannath Temple.

Paul Kelly is a mathematician known for his contributions to various areas of mathematics, particularly in the field of graph theory, combinatorics, and mathematical logic. He has published research papers and is known for his work on topics such as combinatorial designs and algorithms. In addition to his academic work, Paul Kelly may also be involved in teaching and mentoring students in mathematics, contributing to the development of the mathematical community. His influence may extend through publications, conferences, and collaborations with other mathematicians.

Ping Zhang is a mathematician known for his contributions to graph theory, a branch of mathematics that studies graphs, which are mathematical structures used to model pairwise relationships between objects. His research often focuses on topics such as graph coloring, extremal graph theory, and the properties of specific types of graphs. Zhang has published various papers and has been involved in advancing the understanding of different aspects of graph theory.

As of my last update in October 2021, there isn't any widely known public figure, event, or concept specifically called "Regina Tyshkevich." It's possible that this name could refer to a private individual or a less prominent figure not covered in mainstream sources.

V. J. Havel may refer to Václav Havel, a prominent Czech playwright, dissident, and politician who served as the last president of Czechoslovakia from 1989 to 1992 and the first president of the Czech Republic from 1993 to 2003. He was a leading figure in the Velvet Revolution, which peacefully ended communist rule in Czechoslovakia.

"W. G. Brown" could refer to various entities, depending on the context. Common associations with the name might include: 1. **W. G. Brown (Author)**: It could refer to a specific author or writer known for their works. However, there might not be a widely recognized figure with that name. 2. **Business or Brand**: It might be a company or brand, particularly in fields like literature, publishing, or other industries.

Sharadchandra Shankar Shrikhande was an Indian mathematician known for his contributions to various fields in mathematics, particularly in combinatorial designs and finite geometries. He is often recognized for his work on the existence of certain combinatorial structures and the development of the Shrikhande graph, which is a specific graph in graph theory that exhibits interesting properties related to symmetry and structure.

The Nu function is not a standard mathematical or scientific function widely recognized in literature or academia. However, if you are referring to a function or concept that is known by a specific name or acronym, please provide more context.

Stirling's approximation is a formula used to approximate the factorial of a large integer \( n \). It is particularly useful in combinatorics, statistical mechanics, and various areas of mathematics and physics where factorials of large numbers arise. The approximation is given by the formula: \[ n!

The trigamma function, denoted as \(\psi' (x)\) or sometimes as \(\mathrm{Trigamma}(x)\), is the derivative of the digamma function \(\psi(x)\), which is itself the logarithmic derivative of the gamma function \(\Gamma(x)\).

Trigonometry is a branch of mathematics that studies the relationships between the angles and sides of triangles, particularly right-angled triangles. The word "trigonometry" comes from the Greek words "trigonon" (triangle) and "metron" (measure).

The Soboleva modified hyperbolic tangent function, often represented as \( \tanh_s(x) \), is a mathematical function that is a modification of the standard hyperbolic tangent function. In various domains, including physics and engineering, such modified functions are introduced to better handle specific properties such as asymptotic behavior, smoothness, or to meet certain boundary conditions.

Dixon elliptic functions are a set of functions that arise in the theory of elliptic functions, which are complex functions that are periodic in two different directions. Specifically, Dixon elliptic functions are a generalization of the classical elliptic functions and are studied primarily in the context of algebraic functions and complex analysis. Named after the mathematician Alfred William Dixon, these functions have particular properties that make them useful in various branches of mathematics, including number theory, algebraic geometry, and mathematical physics.

The half-period ratio, often referred to in the context of periodic functions, is a mathematical concept that describes the relationship between the periods of a function and its symmetry properties. Specifically, for a periodic function, the half-period ratio relates the half-period to the full period of the function. More formally, if \( T \) is the full period of a periodic function, then the half-period, denoted as \( T/2 \), is simply half of that period.

The bilateral hypergeometric series is a generalization of the ordinary hypergeometric series, which allows for the summation of terms indexed by two parameters rather than one.

The Frobenius solution to the hypergeometric equation refers to the method of finding a series solution near a regular singular point of the hypergeometric differential equation.

Star coloring is a type of graph coloring in which the vertices of a graph are assigned colors such that no two adjacent vertices share the same color and additionally, no two vertices at a distance of two (i.e., connected through a single vertex) have the same color.