Tricomi–Carlitz polynomials are a class of polynomials that arise in the study of $q$-analogues in the context of basic hypergeometric series and combinatorial identities. They are named after the mathematicians Francesco Tricomi and Leonard Carlitz, who studied these polynomials in relation to $q$-series. These polynomials can be defined through various generating functions and properties related to $q$-binomial coefficients.
Wilson polynomials, denoted as \( W_n(x) \), are a class of orthogonal polynomials that arise in the context of probability theory and statistical mechanics. They are defined on the interval \( (0, 1) \) and are associated with the Beta distribution. Wilson polynomials can be expressed using the following formula: \[ W_n(x) = \frac{n!}{(n + 1)!
The 3SUM problem is a classic algorithmic problem in computer science, particularly in the fields of computer algorithms and complexity theory. The problem can be stated as follows: Given an array of integers, the task is to determine if there exist three distinct indices \( i, j, k \) such that the sum of the elements at these indices is equal to zero, i.e.
The Element Distinctness problem is a fundamental problem in computer science and algorithms, particularly in the area of data structures and complexity theory. The problem can be succinctly described as follows: **Problem Statement:** Given a set of \( n \) elements, determine if all the elements are distinct or if there are any duplicates in the set.
In chemistry, "curing" refers to a process in which a material, often a polymer or resin, is hardened or set through a chemical reaction. This process typically involves the addition of a curing agent, heat, or ultraviolet (UV) light to initiate a cross-linking reaction, which transforms the initially soft or liquid material into a solid, durable structure.
Cyclic olefin copolymers (COCs) are a class of thermoplastic polymers that are derived from the polymerization of cyclic olefin monomers. These materials are known for their unique combination of properties, which include high transparency, low moisture absorption, excellent chemical resistance, and good mechanical strength. COCs typically have a low density and can be molded easily into various shapes, making them suitable for a wide range of applications.
The degree of polymerization (DP) is a measure that indicates the number of repeating units in a polymer chain. It is essentially the number of monomeric units that are joined together to form a larger polymer molecule. The DP can provide insights into the properties of the polymer, such as its molecular weight, physical characteristics, and performance in applications.
Dentine bonding agents are specialized materials used in dentistry to bond restorative materials, such as composites, to the dentine layer of the tooth structure. Dentine is the layer beneath the enamel that provides support and structure to the tooth. The bonding of materials to dentine is crucial for the long-term success of dental restorations, as it helps to create a seal that prevents microleakage and enhances the overall durability of the restoration.
The term "Ideal chain" can refer to different concepts depending on the context. Here are a few interpretations: 1. **Supply Chain Management**: In supply chain contexts, an "ideal chain" may refer to a perfectly optimized supply chain that operates with maximum efficiency, minimal waste, and seamless coordination between suppliers, manufacturers, distributors, and retailers.
The Kaminsky catalyst refers to a class of catalysts developed by chemist Nikolai Kaminsky, primarily used in the field of organic synthesis. One of the most notable applications of the Kaminsky catalyst is in the polymerization of olefins, particularly in the context of creating various types of polymers and copolymers. The Kaminsky catalyst usually involves a combination of transition metal compounds and other ligands, which facilitate the polymerization process.
Kinetic chain length refers to the concept that describes the total distance over which forces and movements are applied in a kinetic chain during physical activities. In biomechanics, the kinetic chain is a sequence of segments (typically the joints and limbs) that work together to produce movement. Each segment of the body can be thought of as an individual link in this chain.
The Kuhn length is a concept in polymer physics that describes the effective length of a segment of a polymer chain that behaves as though it is a rigid rod. It is named after the physicist William Kuhn, who contributed to the understanding of polymer behavior. In a simplified model, a polymer chain can be thought of as being composed of many such rigid segments (or "Kuhn segments"), which are connected by flexible linkages.
Ladder polymers are a type of polymeric structure characterized by their unique arrangement, which resembles a ladder. In these materials, the polymer chains are structured with rigid backbones and are connected by side groups or links that form steps in the "ladder." This configuration can lead to distinctive properties, such as high thermal stability, rigidity, and resistance to solvents and chemicals.
The Mark–Houwink equation describes the relationship between the intrinsic viscosity \([η]\) of a polymer solution and the molecular weight \(M\) of the polymer. This empirical relationship is significant in polymer science as it provides insights into the size and shape of macromolecules in solution.
The Mayo–Lewis equation is a relationship used in polymer science to describe the relationship between the glass transition temperature (Tg) of a polymer and its molecular weight. It is particularly relevant when discussing polymers that exhibit glass transition behavior, which is the temperature range below which the polymer becomes brittle and behaves like a glass.
The Hansen solubility parameter (HSP) is a quantitative measure used to predict the solubility of materials, particularly polymers, in different solvents. Developed by Charles M. Hansen in the 1960s, the HSP divides the solubility parameter into three components, each addressing different types of interactions between molecules: 1. **Dispersion Forces (δD)**: This component relates to the van der Waals forces that arise from temporary dipoles in molecules.
Herbert Morawetz is a prominent mathematician known for his contributions to mathematical analysis and partial differential equations. He is particularly recognized for his work in the field of dispersive equations and his impact on areas such as fluid dynamics and wave propagation. His research has advanced the understanding of various mathematical phenomena, and he has been influential in both theoretical developments and applied mathematics.
The Hildebrand solubility parameter is a numerical value that characterizes the solvency properties of a solvent or material. It is part of a broader concept used in polymer science and material science to predict the solubility and compatibility of different materials, particularly polymers with solvents or other polymers.
Hollow fiber membranes are tubular structures made from polymer or ceramic materials that are designed to selectively separate fluids based on certain properties, such as size or charge. These membranes have a large surface area and can be arranged in a dense, compact configuration, making them highly efficient for various applications. **Key Characteristics of Hollow Fiber Membranes:** 1. **Structure**: They consist of thin, hollow fibers with a lumen (inner space) that allows fluids to flow through them.