Contact geometry is a branch of differential geometry that deals with contact manifolds, which are odd-dimensional manifolds equipped with a special kind of geometrical structure called a contact structure. This structure can be thought of as a geometric way of capturing certain properties of systems that exhibit a notion of "direction," and it is closely related to the study of dynamical systems and thermodynamics.

A diffeomorphism is a concept from differential geometry and calculus, representing a special type of mapping between smooth manifolds. Specifically, a diffeomorphism is a function that meets the following criteria: 1. **Smoothness**: The function is infinitely differentiable (i.e., it is a C^∞ function) and its inverse is also infinitely differentiable.

In differential topology, theorems refer to fundamental results that explore the properties and structures of differentiable manifolds and their mappings. This branch of mathematics merges concepts from both differential geometry and algebraic topology, and it investigates how smooth structures behave under various transformations.

"Band sum" isn't a widely recognized term in mathematics or related fields, as of my last update in October 2023. However, the term could possibly be used in various contexts, such as in statistics, computing, or even in specific branches of applied mathematics. If you’ve encountered "band sum" in a particular context, please provide more details.

X-ray scattering is a powerful analytical technique used to study the structural properties of materials at the atomic or molecular level. This method involves directing X-rays at a sample and analyzing the way these rays scatter off the material. The scattering of X-rays can provide valuable information about the arrangement of atoms, molecular structures, phase transitions, and other properties of the sample.

Babinet's principle is a concept in wave optics that relates to the diffraction of waves, particularly light, when they encounter an obstacle or aperture. It states that the diffraction pattern created by a particular aperture is identical to the diffraction pattern produced by an obstacle of the same shape and size but with its area blocked instead of open.

In mathematics, particularly in the field of topology and differential geometry, a "double manifold" typically refers to a space formed by taking two copies of a manifold and gluing them together along a common boundary or a particular subset. However, the term "double manifold" can also refer to other specific constructions depending on the context.

An exotic sphere is a differentiable manifold that is homeomorphic but not diffeomorphic to the standard \( n \)-dimensional sphere \( S^n \) in Euclidean space. This means that while two spaces may have the same topological structure (they can be continuously deformed into each other without tearing or gluing), they have different smooth structures (the way we can differentiate functions on them).

The Kervaire manifold, specifically the Kervaire manifold of dimension \( 2n+1 \) for \( n \geq 1 \), is a type of differentiable manifold that arises in the study of smooth structures on high-dimensional spheres and exotic \( \mathbb{R}^n \). It is named after mathematician Michel Kervaire.

The Inverse Function Theorem is a fundamental result in differential calculus that provides conditions under which a function has a differentiable inverse. It states the following: ### Statement of the Theorem: Let \( f: U \to \mathbb{R}^n \) be a function defined on an open set \( U \subseteq \mathbb{R}^n \). If: 1. \( f \) is continuously differentiable in \( U \), 2.

Milnor's sphere refers to a specific example of a manifold that was discovered by mathematician John Milnor in the 1950s. It is particularly known for being a counterexample in differential topology, specifically in the context of the classification of high-dimensional spheres. In more detail, Milnor constructed a manifold that is homeomorphic (topologically equivalent) to the 7-dimensional sphere \( S^7 \) but not diffeomorphic (smoothly equivalent) to it.

A vector field is a mathematical construct that assigns a vector to every point in a space. It can be thought of as a way to represent spatial variations in a quantity that has both magnitude and direction. Vector fields are widely used in physics and engineering to model phenomena such as fluid flow, electromagnetic fields, and gravitational fields, among others.

A **parallelizable manifold** is a differentiable manifold that has a global frame of vector fields. This means there exists a set of smooth vector fields that span the tangent space at every point of the manifold, and these vector fields can be chosen to vary smoothly. In more formal terms, a manifold \( M \) is said to be parallelizable if there exists a smooth bundle of vector fields \( \{V_1, V_2, ...

The Schoenflies problem is a question in the field of topology, specifically concerning the embedding of spheres in Euclidean spaces. It is named after the mathematician Rudolf Schoenflies. The problem essentially asks whether every simple, closed curve in three-dimensional space can be "filled in" by a disk, or in more technical terms, whether every homeomorphism of the 2-sphere (the surface of a solid ball) can be extended to a homeomorphism of the solid ball itself.

In mathematics, particularly in the field of differential geometry, a **submanifold** is a subset of a manifold that itself has the structure of a manifold, often with respect to the topology and differential structure induced from the larger manifold.

Faà di Bruno's formula provides a way to compute the \( n \)-th derivative of a composed function. It generalizes the chain rule for differentiation in a systematic way.

The term "reciprocal rule" can refer to different concepts depending on the context in which it's used. Below are a few interpretations of the term: 1. **Mathematics**: In mathematics, particularly in the context of fractions or division, the reciprocal of a number is defined as \(1\) divided by that number.

A diffraction grating is an optical component with a periodic structure that diffracts light into several beams. The grating consists of multiple closely spaced slits or grooves, which can be either reflective or transmissive. When light waves encounter the grating, they are scattered at specific angles according to the principles of wave interference, producing a spectrum of colors.

"Discoveries" by Juraj Tóth is not a widely recognized title in mainstream literature, art, or scientific fields up to my last update in October 2023. However, there may be specific niche areas or emerging works that have not gained significant attention yet. Juraj Tóth could also be involved in various domains, such as scientific research, literature, or academia.

CORDIC, which stands for COordinate Rotation DIgital Computer, is an algorithm used for calculating trigonometric functions, hyperbolic functions, exponentials, logarithms, and square roots, among other operations. It was first introduced by Volder in 1959 and has become a popular method for implementing these calculations in hardware, particularly in dedicated digital processors and embedded systems where resources are limited.