Backcoating is a process used in the manufacturing of textiles and various types of materials, typically to enhance durability, moisture resistance, or other functional properties. It involves the application of a layer of material (often a polymer or adhesive) to the back side of a fabric or a substrate. This backing layer can provide several benefits: 1. **Increased Durability:** The backcoating can reinforce the base material, making it more resistant to wear and tear.

The Askey–Gasper inequality is a result in the field of mathematical analysis, particularly in the study of special functions and orthogonal polynomials. It provides bounds for certain types of sums and integrals involving orthogonal polynomials, especially within the context of Jacobi polynomials.

Discrete Chebyshev polynomials are a sequence of orthogonal polynomials defined on a discrete set of points, typically related to the Chebyshev polynomials of the first kind. These discrete polynomials arise in various applications, including numerical analysis, approximation theory, and computing discrete Fourier transforms. The discrete Chebyshev polynomials are defined based on the characteristic roots of the Chebyshev polynomials, which correspond to specific points on an interval.

"Paper Planes" is a song by the British rapper M.I.A., released in 2008 as part of her album "Kala." The song became widely popular for its catchy chorus, which features the iconic sound of cash registers and gunshots, symbolizing themes of capitalism and violence. "Paper Planes" received critical acclaim and commercial success, charting in multiple countries and becoming a cultural touchstone.

A crease pattern is a specific arrangement of folds typically used in origami, paper engineering, and other forms of foldable structures. It is a two-dimensional diagram that represents all the creases needed to be made in a sheet of paper to create a particular three-dimensional shape or object. Crease patterns are often represented visually, showing both mountain folds (where the paper is folded upwards) and valley folds (where the paper is folded downwards).

The concept of "one thousand origami cranes," or "Senbazuru" in Japanese, is a significant cultural symbol in Japan. According to Japanese legend, if someone folds one thousand origami cranes, they will be granted a wish, often interpreted as the wish for good health, long life, or even world peace. The tradition is especially associated with Sadako Sasaki, a young girl who became a victim of the Hiroshima atomic bombing.

"Sonobe" typically refers to a geometric construction technique associated with modular origami, which involves assembling unit blocks to create complex three-dimensional structures. The Sonobe unit is a specific polygon, usually made from a square piece of paper, that can be folded and assembled with other Sonobe units to form various polyhedral shapes. The Sonobe unit is comprised of a square that is folded into a specific pattern, allowing it to interlock with other units without the use of adhesive.

The Yoshizawa–Randlett system is a mathematical framework used to model and analyze certain types of dynamical systems, particularly in the context of nonlinear dynamics and chaos theory. This system is named after the researchers Yoshizawa and Randlett, who contributed to the study of systems that exhibit complex behavior under specific conditions.

Associated Legendre polynomials are a generalization of Legendre polynomials, which arise in the context of solving problems in physics, particularly in potential theory, quantum mechanics, and in the theory of spherical harmonics. The associated Legendre polynomials, denoted as \( P_\ell^m(x) \), are defined for non-negative integers \( \ell \) and \( m \), where \( m \) can take on values from \( 0 \) to \( \ell \).

Orthogonal polynomials on the unit circle are a class of polynomials that are orthogonal with respect to a specific inner product defined on the unit circle in the complex plane. These polynomials have important applications in various fields, including approximation theory, numerical analysis, and spectral theory.

Plancherel–Rotach asymptotics refers to a set of results in the asymptotic analysis of certain special functions and combinatorial quantities, particularly associated with orthogonal polynomials and probability distributions. The results originally emerged from studying the asymptotic behavior of the zeros of orthogonal polynomials, and they have applications in various areas, including statistical mechanics, random matrix theory, and combinatorial enumeration.

The "Jack function" (also known as the Jack polynomial) is a type of symmetric polynomial that generalizes the Schur polynomials. Jack polynomials depend on a parameter \( \alpha \) and are indexed by partitions. They can be used in various areas of mathematics, including combinatorics, representation theory, and algebraic geometry.

Little \( q \)-Laguerre polynomials are a family of orthogonal polynomials that arise in the context of \( q \)-calculus, which is a generalization of classical calculus. They are particularly important in various areas of mathematics and mathematical physics, including combinatorics, special functions, and representation theory.

Yuliya Mishura, sometimes spelled Yulia Mishura, may refer to a person involved in various fields, but without more specific context, it's difficult to provide accurate information.

The Q-Charlier polynomials are a family of orthogonal polynomials that arise in the context of probability and combinatorial analysis. They are a specific case of the Charlier polynomials, which are defined concerning Poisson distribution. The Q-Charlier polynomials extend this concept to the setting of the \( q \)-calculus, which incorporates a parameter \( q \) that allows for generalization and flexibility in combinatorial structures.

The Q-Krawtchouk polynomials are a set of orthogonal polynomials that generalize the Krawtchouk polynomials, which themselves are a class of discrete orthogonal polynomials. The Krawtchouk polynomials arise in combinatorial settings and are connected to binomial distributions, while the Q-Krawtchouk polynomials introduce a parameter \( q \) that allows for further generalization. ### Definition and Properties 1.

Quantum \( q \)-Krawtchouk polynomials are a family of orthogonal polynomials that can be seen as a \( q \)-analogue of the classical Krawtchouk polynomials. They arise in various areas of mathematics, particularly in the theory of quantum groups, representation theory, and combinatorial analysis. ### Definitions and Properties 1.

The term "perpendicular" refers to the relationship between two lines, segments, or planes that meet or intersect at a right angle (90 degrees). In two-dimensional geometry, if line segment \( AB \) is perpendicular to line segment \( CD \), it means they intersect at an angle of 90 degrees. In three-dimensional space, the concept extends similarly; for example, a line can be said to be perpendicular to a plane if it intersects the plane at a right angle.

Bin packing is a type of combinatorial optimization problem that involves packing a set of items of varying sizes into a finite number of bins or containers in such a way that minimizes the number of bins used. The objective is to efficiently utilize space (or capacity) while ensuring that the items fit within the constraints of the bins. ### Key Concepts 1. **Items**: Each item has a specific size or weight. 2. **Bins**: Each bin has a maximum capacity that cannot be exceeded.

In geometry, the term "normal" can refer to several concepts, but it is most commonly used in relation to the idea of a line or vector that is perpendicular to a surface or another line. Here are a few contexts in which "normal" is used: 1. **Normal Vector:** In three-dimensional space, a normal vector to a surface at a given point is a vector that is perpendicular to the tangent plane of the surface at that point.