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.

OS/2, short for Operating System/2, is an operating system developed by IBM in collaboration with Microsoft in the late 1980s. It was initially designed as a successor to DOS and intended to be a more advanced platform for personal computing. The OS/2 operating system featured a graphical user interface and was known for its multitasking capabilities, stability, and support for running multiple applications simultaneously. OS/2 went through several versions, with notable releases including OS/2 1.

The 20th century saw significant contributions from Pakistani mathematicians, particularly in the context of the country's formation and its growth in higher education and research. Here are some notable mathematicians from Pakistan during that time: 1. **Abdul Salam**: Although primarily known for his work in theoretical physics, Abdul Salam also made significant contributions to mathematical physics. He was awarded the Nobel Prize in Physics in 1979. 2. **Muhammad G.

Presentation Manager, often associated with IBM's OS/2 operating system, is a graphical user interface (GUI) environment that enables users to create, manage, and present information in a visually appealing manner. This software provides tools for developing presentations, including slides, graphics, and multimedia elements, similar to applications like Microsoft PowerPoint.

Astronomy images are photographs or visual representations of celestial objects and phenomena captured through telescopes, cameras, and other imaging equipment. These images can include a wide range of subjects, such as: 1. **Planets**: Photographs of planets in our solar system, showing their surfaces, atmospheres, and moons. 2. **Stars**: Images of individual stars or groups of stars, including their colors, brightness, and formation.

Songs about outer space often explore themes of exploration, wonder, and existential reflection. They may delve into the vastness of the universe, the idea of life on other planets, or the human experience in relation to the cosmos. Here are some notable examples across various genres: 1. **"Rocket Man" by Elton John** - A poignant reflection on the loneliness and isolation of a space traveler.