Upset welding is a type of resistance welding process used to join two metal parts together by generating heat through the resistance of the materials. In upset welding, two workpieces are brought together under pressure. An electric current is passed through the interface of the materials, causing localized heating at the contact point due to electrical resistance. Once the materials reach their melting temperature, they are upset (compressed) further to create a solid bond as the molten area cools and solidifies.
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.
Q-Meixner polynomials are a class of orthogonal polynomials that generalize the classical Meixner polynomials. They are typically associated with specific probability distributions, particularly in the context of q-calculus, which is a branch of mathematics dealing with q-series and q-orthogonal polynomials. Meixner polynomials arise in probability theory, especially in relation to certain types of random walks and discrete distributions.
The Q-Meixner–Pollaczek polynomials are a family of orthogonal polynomials that arise in the context of certain special functions and quantum mechanics. They are a generalization of both the Meixner and Pollaczek polynomials and are associated with q-analogues, which are modifications of classic mathematical structures that depend on a parameter \( q \).
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.
Rogers polynomials are a family of orthogonal polynomials that arise in the context of approximation theory and special functions. They are closely related to the theory of orthogonal polynomials on the unit circle and have connections to various areas of mathematics, including combinatorics and number theory.
Sobolev orthogonal polynomials are a generalization of classical orthogonal polynomials that arise in the context of Sobolev spaces. In classical approximation theory, orthogonal polynomials, such as Legendre, Hermite, and Laguerre polynomials, are orthogonal with respect to a weight function over a given interval or domain. Sobolev orthogonal polynomials extend this concept by introducing a notion of orthogonality that involves both a weight function and derivatives.
Zernike polynomials are a set of orthogonal polynomials defined over a unit disk, which are commonly used in various fields such as optics, imaging science, and surface metrology. They are particularly useful for describing wavefronts and optical aberrations, as they provide a convenient mathematical framework for representing complex shapes and patterns.
Orthogonal coordinate systems are systems used to define a point in space using coordinates in such a way that the coordinate axes are perpendicular (orthogonal) to each other. In these systems, the position of a point is determined by a set of values, typically referred to as coordinates, which indicates its distance from the axes.
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.
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.
Perpendicular distance refers to the shortest distance from a point to a line, plane, or a geometric shape. This distance is measured along a line that is perpendicular (at a 90-degree angle) to the surface or line in question. ### Key Points: - **From a Point to a Line**: The perpendicular distance from a point to a line is the length of the segment that connects the point to the line at a right angle.
Thomas Jones was a mathematician primarily known for his contributions to the field of mathematics education and his work related to number theory. While there are several individuals with the same name, the most notable Thomas Jones in mathematics is often recognized for his writings and efforts in promoting mathematical understanding, particularly at a time when the field was evolving rapidly. He may not be as widely recognized as some of his contemporaries, but his influence on mathematics education has been acknowledged in academic circles.
OS/2, which stands for Operating System/2, is a computer operating system developed by IBM in the late 1980s. Originally created as a successor to DOS, OS/2 was intended to provide a stable and capable environment for running applications in a multitasking and multiuser setting. It was co-developed with Microsoft initially, but after version 1.3, IBM took over development completely.
EComStation is an operating system that is based on IBM's OS/2, developed by Serenity Systems International and later by other groups. It aims to provide a modernized platform for users who appreciate the unique features of OS/2, while also offering support for newer hardware and software. EComStation includes a graphical user interface, support for multitasking, and compatibility with various applications that were originally designed for OS/2.
IBM LAN Server is a networking software product that was developed by IBM to enable file and print sharing, as well as other networking functions, within local area networks (LANs). It was originally designed for use with IBM OS/2 operating systems and later supported Windows and other operating systems. Key features of IBM LAN Server included: 1. **File and Print Sharing**: It facilitated the sharing of files and printers among multiple users in a networked environment.
The IBM PS/2 (Personal System/2) was a line of personal computers introduced by IBM in April 1987. It was designed to succeed the IBM PC and PC/AT lines, offering advancements in hardware and software compatibility. The PS/2 line was significant for several reasons: 1. **Microchannel Architecture (MCA)**: PS/2 introduced the Microchannel Architecture, a new bus standard that replaced the older ISA (Industry Standard Architecture) used in previous IBM PCs.
An Installable File System (IFS) is a type of file system architecture that allows users to add new file system types or formats to an operating system without requiring changes to the core system itself. This is typically accomplished through a plugin or module system, where new file systems can be installed as additional components. ### Key Features of Installable File Systems: 1. **Modularity**: IFS provides a modular approach to file systems.