The British Tabulating Machine Company (BTM) was a British firm primarily known for its role in the development and manufacture of tabulating and computing equipment in the early to mid-20th century. Established in 1896, BTM specialized in creating devices that utilized punched cards for data processing, a technology that was widely used for statistical calculations and data management before the advent of electronic computing. BTM played a significant role in the introduction and implementation of automatic data processing systems in the UK.

Robert Morris is a mathematician known for his work in various areas of mathematics, particularly in number theory and combinatorics. He is also well-known for his contributions to the field of mathematical logic. Morris has made significant contributions to the understanding of mathematical structures and has published numerous papers and articles in respected mathematical journals. One notable aspect of his work is his involvement in the study of mathematical problems and algorithms, as well as his exploration of the connections between different areas of mathematics.

Sergey Kitaev is a prominent theoretical physicist and mathematician known for his work in quantum computation, condensed matter physics, and mathematical physics. He made significant contributions to the development of topological quantum computation, particularly through his work on anyons and topological phases of matter. Kitaev is also well-known for the Kitaev model, which describes a lattice model of spinless fermions that provides insights into the behavior of quantum systems and is relevant to quantum computing.

Vojtěch Rödl is a prominent Czech mathematician known for his work in combinatorics, graph theory, and theoretical computer science. He has made significant contributions to various areas of mathematics, particularly in the study of random structures and extremal combinatorics. Rödl is also known for the Rödl's theorem, which is a result in extremal combinatorics. Throughout his career, he has published numerous papers and has been involved in mathematical education and research.

Zdeněk Hedrlín is a Czech diplomat known for his contributions to international relations and diplomacy. Specific details about his career, roles, and achievements may not be widely publicized, but he has been involved in various diplomatic positions representing the Czech Republic.

In combinatorics, a "necklace" is a mathematical object that represents a circular arrangement of beads (or other distinguishing objects) where rotations and reflections are considered equivalent. Necklaces can be used to model problems involving the arrangement of identical or distinct objects in a way that takes into account the symmetry of the arrangement. ### Key Points about Necklaces: 1. **Rotational Symmetry**: A necklace can be rotated, and arrangements that are rotations of one another are considered identical.

Word problems for groups typically involve scenarios where you need to solve for quantities related to a group of items or individuals. They often require understanding relationships between the items or people in the group, applying mathematical concepts such as addition, subtraction, multiplication, or division. Here are a few examples: ### Example 1: Classrooms **Problem:** In a school, there are 3 classrooms. Each classroom has 24 students.

The Landau–Lifshitz model, often referred to in the context of magnetism, specifically deals with the theoretical description of magnetization dynamics in ferromagnetic materials. It is named after the physicists Lev Landau and Emil Lifshitz, who contributed significantly to the field of theoretical physics. The model primarily provides a framework to describe the evolution of the magnetization vector \(\mathbf{M}\) in a ferromagnet.

The Canadian Traveller Problem (CTP) is a combinatorial optimization problem that extends the classic Travelling Salesman Problem (TSP). It arises in scenarios where a traveller must visit a set of locations (cities or nodes) while adhering to certain constraints.

Graph partitioning is a technique in computer science and mathematics that involves dividing a graph into smaller, disjoint subgraphs or partitions, such that certain criteria are optimized. The graph typically consists of vertices (or nodes) and edges (which connect the vertices).

The Graph Realization Problem is a well-studied problem in graph theory and combinatorial optimization. It involves determining whether a given graph can be realized as the intersection graph of a set of geometric objects, such as points, lines, circles, or polygons, in a specific dimension or space.

The Maximum Agreement Subtree (MAST) problem is a computational problem in the field of comparative genomics and bioinformatics. It involves identifying a subtree that is common to multiple phylogenetic trees (or evolutionary trees) that represent the relationships between a given set of species or taxa. Specifically, the goal is to find a subtree that maximizes the number of leaves (species) that are consistent across the input trees.

Quadratic pseudo-Boolean optimization refers to the optimization of a specific type of mathematical function known as a quadratic pseudo-Boolean function. These functions are special cases of polynomial functions and are defined over binary variables (typically taking values of 0 or 1).

The Set Traveling Salesman Problem (Set TSP) is a variant of the classic Traveling Salesman Problem (TSP), which is a well-known problem in combinatorial optimization. In the standard TSP, a salesman is required to visit a set of cities exactly once and return to the starting point while minimizing the total distance traveled.

A star system, often referred to as a stellar system, is a group of celestial bodies that are gravitationally bound to a central star. The most recognizable type of star system is a solar system, which includes a star (or multiple stars in the case of binary or multiple star systems) and various objects such as planets, moons, asteroids, comets, and meteoroids that orbit the star.

Bifurcation theory, a branch of mathematics and dynamical systems, studies how the qualitative or topological structure of a given system changes as parameters vary. This theory has several biological applications across various fields. Here are some notable ones: 1. **Population Dynamics**: Bifurcation theory is often used to model changes in population dynamics of species in ecological systems.

The Bogdanov–Takens bifurcation is a significant phenomenon in the study of dynamical systems, particularly in the context of the behavior of nonlinear systems. It describes a scenario in which a system undergoes a bifurcation, leading to the simultaneous occurrence of a transcritical bifurcation (where the stability of fixed points is exchanged) and a Hopf bifurcation (where a fixed point becomes unstable and bifurcates into a periodic orbit).

Pitchfork bifurcation is a type of bifurcation that occurs in dynamical systems, particularly in the study of nonlinear systems. It describes a situation where a system's stable equilibrium point becomes unstable and gives rise to two new stable equilibrium points as a parameter is varied. In more technical terms, a pitchfork bifurcation typically occurs in systems described by equations where the steady-state solutions undergo a change in stability.

The Horseshoe map is a well-known example of a one-dimensional dynamical system that exhibits chaotic behavior. It is a type of chaotic map that is used in the study of chaos theory and nonlinear dynamics. The Horseshoe map illustrates how simple deterministic systems can exhibit complex, unpredictable behavior. ### Definition The Horseshoe map can be defined on the unit interval \( [0, 1] \) and involves a transformation that stretches and folds the interval to create a "horseshoe" shape.