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 \( \ell \).
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.
The Askey scheme is a classification of orthogonal polynomial sequences that arise in the context of special functions and approximation theory. Named after Richard Askey, this scheme organizes orthogonal polynomials into a hierarchy based on their properties and relationships.
The Al-Salam–Chihara polynomials are a family of orthogonal polynomials that arise in the theory of special functions, specifically in the context of q-series and quantum calculus. They are named after the mathematicians Abd al-Rahman Al-Salam and Jun-iti Chihara, who contributed to their study.
Al-Salam–Carlitz polynomials are a family of orthogonal polynomials that generalize the classical Carlitz polynomials. They appear in the context of q-series and combinatorial identities and are related to various areas in mathematics, including number theory and formal power series. These polynomials are typically defined in terms of parameters \( a \) and \( b \) and a variable \( x \).
An affine root system is an extension of the concept of root systems, which are used in the theory of Lie algebras and algebraic groups. The affine root system is associated with affine Lie algebras, which are a class of infinite-dimensional Lie algebras that arise in the study of symmetries and integrable systems.
Affine \( q \)-Krawtchouk polynomials are a family of orthogonal polynomials that arise in the context of quantum calculus or non-classical orthogonal polynomial theory, particularly in relation to \( q \)-analogs of established mathematical concepts. These polynomials generalize the classical Krawtchouk polynomials, which are associated with the binomial distribution and combinatorial problems.
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.
Wet-folding is a technique used in origami that involves moistening the paper before folding it. This method allows the paper to become more pliable and easier to manipulate, which enables the folder to achieve smoother curves and more intricate shapes that might be difficult to create with dry paper. The key benefits of wet-folding include: 1. **Enhanced Sculptural Quality**: By using moisture, the paper can hold more complex and rounded forms, resulting in a more organic appearance.
Washi is a traditional Japanese paper known for its unique texture, strength, and versatility. It is made from the fibers of plants such as the gampi tree, the mitsumata shrub, or the paper mulberry. The production of washi involves a labor-intensive process that includes hand-pulping and hand-pouring the paper, resulting in a product that is both highly decorative and functional.
Troublewit is a type of paper craft made from a single strip of paper that is folded in a specific way to create a three-dimensional object, typically resembling a shape that can be manipulated. It's often used as a classic children's toy or a decorative item. The folding technique can create interesting visual effects and movements as the paper is twisted and turned. The term "troublewit" can also refer to the specific object created through this method.
Tamatebako, often referred to as the "jewel box" or "treasure box," is a traditional Japanese origami design. It is known for its beautiful and intricate folding technique, allowing the paper to create a 3D box that can be opened. The design is typically made from a single square piece of paper and is folded in such a way that it can hold small items, resembling a treasure trove or a decorative container.
"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.
Shide is a traditional Japanese ritual paper streamer that plays a significant role in Shinto practices. It is typically made from white paper or rapeseed and is characterized by its zigzag or folded shape. Shide is often used as a symbol of purity and to ward off evil spirits. In Shinto shrines, shide can be found hanging from sacred objects or attached to torii gates, marking areas considered sacred.
Rigid origami is a branch of origami that focuses on the folding of flat materials into three-dimensional shapes while maintaining the rigidity of the folded structure. Unlike traditional origami, where paper can be flexibly bent and manipulated, rigid origami relies on specific types of folds that allow a structure to maintain its shape without any deformation.
Pureland origami is a style of origami that emphasizes folds that can be made using only straight valley and mountain folds while avoiding complex techniques such as reverse folds, twist folds, and many other advanced techniques. This approach is designed to make origami more accessible, especially for beginners or those with physical limitations. In Pureland origami, the instructions are typically clear and straightforward, using simple terminology and notations.
A paper plane, also known as a paper airplane, is a toy or model airplane made from folded paper. It is created by folding a sheet of paper into a specific shape that allows it to glide through the air when thrown. The design of a paper plane can significantly affect its flying distance, stability, and maneuverability. Making a paper plane typically involves a few simple folds, and there are many different designs that enthusiasts use to achieve various flying characteristics.
A paper fortune teller, also known as a cootie catcher, is a simple origami toy made from a square piece of paper that is manipulated by folding it in a particular way. It consists of four flaps that can be opened and closed, and it is typically used for entertainment and light-hearted fortune-telling. To use a paper fortune teller, a person usually follows these steps: 1. **Create the Paper Fortune Teller**: - Start with a square piece of paper.
Orizuru, often referred to as "folded crane" in Japanese, is a traditional origami figure that symbolizes peace, hope, and healing. The crane has significant cultural importance in Japan and is linked to various legends, one of the most famous being the story of Sadako Sasaki, a Japanese girl who developed leukemia after the atomic bombing of Hiroshima. According to the legend, if someone folds one thousand origami cranes, they will be granted a wish or recover from illness.
Origami paper is a specialized type of paper designed specifically for the art of origami, the Japanese practice of folding paper into intricate shapes and figures. Here are some key characteristics and features of origami paper: 1. **Weight and Thickness**: Origami paper is typically lighter than standard paper, ranging from about 30 to 80 gsm (grams per square meter). This makes it easier to fold and manipulate without tearing.