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Several sets of orthogonal functions have become standard bases for approximating functions. For example, the sine functions sin nx and sin mx are orthogonal on the interval x ∈ ( − π , π ) {\displaystyle x\in (-\pi ,\pi )} when m ≠ n {\displaystyle m\neq n} and n and m are positive integers.
The line segments AB and CD are orthogonal to each other. In mathematics, orthogonality is the generalization of the geometric notion of perpendicularity.Whereas perpendicular is typically followed by to when relating two lines to one another (e.g., "line A is perpendicular to line B"), [1] orthogonal is commonly used without to (e.g., "orthogonal lines A and B").
A simple machine is a mechanical device that changes the direction or magnitude of a force. [1] In general, they can be defined as the simplest mechanisms that use mechanical advantage (also called leverage) to multiply force. [2] Usually the term refers to the six classical simple machines that were defined by Renaissance scientists: [3] [4 ...
An Introduction to Orthogonal Polynomials. Gordon and Breach, New York. ISBN 0-677-04150-0. Chihara, Theodore Seio (2001). "45 years of orthogonal polynomials: a view from the wings". Proceedings of the Fifth International Symposium on Orthogonal Polynomials, Special Functions and their Applications (Patras, 1999).
An example is the pair of sets of respectively left and right eigenvectors of a matrix, indexed by eigenvalue, if the eigenvalues are distinct. [1] A biorthogonal system in which = and ~ = ~ is an orthonormal system.
But unlike the sine and cosine functions, which are continuous, Walsh functions are piecewise constant. They take the values −1 and +1 only, on sub-intervals defined by dyadic fractions. The system of Walsh functions is known as the Walsh system. It is an extension of the Rademacher system of orthogonal functions. [2]
We say that functions and are orthogonal if their inner product (equivalently, the value of this integral) is zero: f , g w = 0. {\displaystyle \langle f,g\rangle _{w}=0.} Orthogonality of two functions with respect to one inner product does not imply orthogonality with respect to another inner product.
The Staudinger ligation is a reaction developed by the Bertozzi group in 2000 that is based on the classic Staudinger reaction of azides with triarylphosphines. [15] It launched the field of bioorthogonal chemistry as the first reaction with completely abiotic functional groups although it is no longer as widely used.