The "Happy Ending Problem" is a classic problem in combinatorial geometry that involves points in a plane. Specifically, it refers to the question of whether a set of points in the plane can be connected to form a convex polygon, and it is typically framed in the context of points positioned in general position (i.e., no three points are collinear).

Szemerédi's theorem is a fundamental result in combinatorial number theory which pertains to arithmetic progressions in sets of integers. Specifically, the theorem states that for any positive integer \( k \), any subset of the integers with positive density contains a non-trivial arithmetic progression of length \( k \). More formally, if \( A \) is a subset of the positive integers with positive upper density, i.e.

Clausen's formula, named after the mathematician Carl Friedrich Gauss and further developed by the German mathematician Karl Clausen, is a formula related to the sums of powers of integers, particularly relevant in number theory and combinatorics. More specifically, Clausen's formula provides a means to express sums of powers of integers in terms of Bernoulli numbers.

The Debye function is a mathematical function that arises in the study of thermal properties of solids, particularly in the context of specific heat and phonon statistics. It is named after the physicist Peter Debye, who introduced it in the early 20th century as part of his work on heat capacity in crystalline solids. The Debye function is used to describe the contribution of phonons (quantized modes of vibrations) to the heat capacity of a solid at low temperatures.

An entire function is a complex function that is holomorphic (i.e., complex differentiable) at all points in the complex plane. In simpler terms, an entire function is a function that can be represented by a power series that converges everywhere in the complex plane. ### Characteristics of Entire Functions: 1. **Holomorphic Everywhere**: Entire functions are differentiable in the complex sense at every point in the complex plane.

The Griewank function is a commonly used test function in optimization and is particularly known for its challenging properties, making it suitable for evaluating optimization algorithms.

The Herglotz–Zagier function is a complex analytic function that arises in the context of number theory and several areas of mathematical analysis. This function is typically expressed in terms of an infinite series and is significant due to its properties related to modular forms and other areas of mathematical research.

The Lamé functions are special functions that arise as solutions to Lamé's differential equation, which is a second-order linear differential equation associated with the problem of a particle constrained to move on an ellipsoid.

A modular form is a complex function that has certain transformation properties and satisfies specific conditions.

The \(\text{sinhc}\) function, often represented as \(\text{sinhc}(x)\), is defined mathematically as: \[ \text{sinhc}(x) = \frac{\sinh(x)}{x} \] for \(x \neq 0\), and \(\text{sinhc}(0) = 1\).

A trigonometric integral is a type of integral that involves trigonometric functions such as sine (sin), cosine (cos), tangent (tan), and their reciprocals or inverses. These integrals often arise in a variety of contexts, including physics, engineering, and mathematics, particularly in calculus when dealing with periodic functions or problems involving angles.

Walsh functions are a set of orthogonal functions that are used in various fields, including signal processing, communications, and computer science. They are defined over the interval [0, 1] and can be extended to other intervals or dimensions. Walsh functions are particularly known for their simplicity and can be represented in a binary form.

Dyson's transform, also known as the Dyson series, is a mathematical tool used primarily in quantum mechanics and quantum field theory. It provides a way to express the time evolution of a quantum state in terms of the interaction Hamiltonian when the system is subject to a time-dependent potential or interaction.

Mnëv's universality theorem is a result in the field of mathematical logic and combinatorial geometry, specifically relating to the arrangement and properties of arrangements of points in the projective plane. It asserts that certain geometric configurations can be used to describe and encode a broad class of mathematical structures. The theorem indicates that the space of geometric configurations — particularly those involving points and lines in a projective space — is rich enough to capture the complexity of various combinatorial and algebraic structures.

The Szemerédi–Trotter theorem is a fundamental result in combinatorial geometry that provides bounds on the incidences between points and lines in the plane. Specifically, it addresses how many points lie on a set of lines, providing a relationship between three parameters: the number of points, the number of lines, and the number of incidences (that is, points that lie on those lines).

Lattice points are points in a coordinate system whose coordinates are all integers. In a two-dimensional Cartesian coordinate system, a lattice point can be represented as \((x, y)\), where both \(x\) and \(y\) are integers. For example, the points \((1, 2)\), \((-3, 4)\), and \((0, 0)\) are all lattice points.

A "2-ring" can refer to different concepts depending on the context, but without specific detail, it's hard to determine exactly what you're asking about. Here are a few possible interpretations: 1. **Mathematics/Abstract Algebra**: In the context of mathematics, particularly in abstract algebra, a "2-ring" might refer to a ring with a specific property or structure; however, this is not a standard term in mathematics.

The term "quadric" typically refers to a specific type of surface or equation in mathematics, particularly in the field of algebraic geometry and analytic geometry.

The Rado covering problem is a classic problem in combinatorics, particularly in the area of graph theory and set theory. The problem is named after mathematician Georgy Rado and deals with the concept of partitioning and covering subsets of sets. The problem can be stated in the following way: You are given a set \( S \), which is typically infinite, and a family of subsets of \( S \).

The Erdős distinct distances problem, posed by the Hungarian mathematician Paul Erdős in 1946, is a question in combinatorial geometry that seeks to determine the minimum number of distinct distances between points in a given finite set in the plane. Specifically, the problem asks for the largest number of points \( n \) that can be placed in the plane such that the number of distinct distances between pairs of points is minimized.