China
Africa
Explore chapters and articles related to this topic
Some Considerations for the use of Orthogonal Transformations on Particulate Morphological Data
Published in John Keith Beddow, T. P. Meloy, Advanced Particulate Morphology, 1980
From the orthonormality and after some manipulations which involve the orthogonality principle, () ∫I[f(t)−f^N0(t)]{Σk=0N−1βk*ϕk(t)}dt=0
Fluid dynamics and wave-structure interactions
Published in Dezhi Ning, Boyin Ding, Modelling and Optimization of Wave Energy Converters, 2022
Malin Göteman, Robert Mayon, Yingyi Liu, Siming Zheng, Rongquan Wang
where the orthonormality of trigonometric functions was used, implying that the only contribution from the product of the cosine functions is when m=n. By comparing (2.23) and (2.27), the conclusion can be drawn that the energy of the nth wave component is given in terms of the amplitude of the wave component as En=12ρgAn2.
Transform methods
Published in John P. D’Angelo, Linear and Complex Analysis for Applications, 2017
The Kronecker delta δjk is a convenient notation when working with orthonormality. The symbol δjk equals 1 when j = k and equals 0 otherwise. Thus orthonormality can be expressed by saying that 〈ϕm,ϕn〉=δmn.
Visualizing Laguerre polynomials as a complete orthonormal set for the inner product space ℙ n
Published in International Journal of Mathematical Education in Science and Technology, 2023
Named after the French mathematician Edmond Laguerre (1834–1886), the set forms an orthogonal set of polynomials with the associated integral inner product . In quantum mechanics, they appear in eigenfunctions satisfying the radial Schrödinger equation for the Hydrogen atom. This note illustrates the integral inner product in the vector space of polynomials of degree with real coefficients – the mapping defined via for – in a DGS/MATLAB-facilitated learning environment. Also explored are the defining properties (symmetry, linearity, positive definiteness) of the integral inner product along with other notions inherent in the inner product space, such as norm, distance, orthogonal projection, Cauchy–Schwarz Inequality, Triangle Inequality, Pythagorean Theorem, Parallelogram Law, orthogonality and orthonormality, orthonormal basis, and coordinates relative to an orthonormal basis. The article also demonstrates the diversity of ways through which Laguerre polynomials can be used as an orthonormal basis for the inner product space in a technology-assisted learning environment. More details on Laguerre Polynomials can be found in Arfken (1985), Hochstrasser (1972), Szegö (1975).
Special affine multiresolution analysis and the construction of orthonormal wavelets in L 2(ℝ)
Published in Applicable Analysis, 2023
Firdous A. Shah, Waseem Z. Lone
Let us start with a function such that and is orthonormal in . Then, we define Conditions (69) and the orthonormality of the system are necessary and sufficient to guarantee that is an orthonormal basis in each , and hence, satisfy the increasing property , for all . Moreover, it follows that satisfy scaling and translating properties of Definition 4.1 also. In order to verify that the ladder of spaces generated by ϕ constitute a special affine MRA, it is sufficient to show that the following properties also hold: Before we prove the main results, we have the following characterization of Riesz basis.
Visualizing the inner product space ℝm×n in a MATLAB-assisted linear algebra classroom
Published in International Journal of Mathematical Education in Science and Technology, 2018
Prelude: This classroom note considers Frobenius inner product – the mapping defined via for real m × n matrices A and B – in a MATLAB-facilitated learning environment. The examples illustrate the defining properties (symmetry, linearity, positive definiteness) of the inner product along with other notions inherent in the inner product space , such as Frobenius norm (length), distance between vectors, angle between vectors, orthogonal projection, Cauchy–Schwarz inequality, triangle inequality, Pythagorean theorem, parallelogram law, orthogonality and orthonormality, orthonormal basis, completeness relation, coordinates relative to an orthonormal basis and Parseval's identity.