A high fidelity simulation is preferred for its remarkable accuracy for engineering problems. However, it requires long computational time, which leads to significant overhead and eventually hinders its application to design study. To overcome such impediments, a reduced-order method is utilized. Reducedorder method can effectively represent a simulation output as a linear combination of a basi...

Moments and transforms are the scalar quantities that characterise a function and capture its significant properties. More specifically, these are the descriptors that correspond to the projection of the function on a specific basis function, where the type or characteristic of the basis function gives the name to the moment or transform [82]. Normally, moments and transforms are treated separa...

A monomial orthogonal polynomial of several variables is of the form x−Qα(x) for a multiindex α ∈ N 0 and it has the least L2 norm among all polynomials of the form xα − P (x), where P and Qα are polynomials of degree less than the total degree of xα. We study monomial orthogonal polynomials with respect to the weight function ∏d+1 i=1 |xi| 2κi on the unit sphere Sd as well as for the related w...

We consider the space Pn of orthogonal polynomials of degree n on the unit disc for a general radially symmetricweight function.We show that there exists a single orthogonal polynomialwhose rotations through the angles j n+1 , j = 0, 1, . . . , n forms an orthonormal basis for Pn, and compute all such polynomials explicitly. This generalises the orthonormal basis of Logan and Shepp for the Lege...

Exercise 1 Suppose that H = C3 and let V be the one-dimensional subspace spanned by |ψy = ?1/3(|0y+ ei/3|1y |2y). Provide an orthonormal basis for VK. Solution: We need to identify a pair of orthonormal states that are also orthogonal to |ψy. An easy first state to take is |ψK 1 y = 1 2 (|0y ei/3|1y). Any other vector of the form a(|0y+ ei/3|1y+ b|2ywill be orthogonal to |ψK 1 y. To also make i...

We construct Jacobi-weighted orthogonal polynomials (α,β,γ) n,r (u,v,w), α,β,γ > −1, α+ β + γ = 0, on the triangular domain T . We show that these polynomials (α,β,γ) n,r (u, v,w) over the triangular domain T satisfy the following properties: (α,β,γ) n,r (u,v,w) ∈ n, n≥ 1, r = 0,1, . . . ,n, and (α,β,γ) n,r (u,v,w) ⊥ (α,β,γ) n,s (u,v,w) for r =s. Hence, (α,β,γ) n,r (u,v,w), n= 0,1,2, . . ., r =...

We introduce a high-order time-domain discontinuous spectral element method for the study of the optical coupling by evanescent whispering gallery modes between two microcylinders, the building blocks of coupled resonator optical waveguide devices. By using the discontinuous spectral element method with a Dubiner orthogonal polynomial basis on triangles and a Legendre nodal orthogonal basis on ...

Abstract. The Calogero model is a one-dimensional quantum integrable system with inverse-square long-range interactions confined in an external harmonic well. It shares the same algebraic structure with the Sutherland model, which is also a one-dimensional quantum integrable system with inverse-sine-square interactions. Inspired by the Rodrigues formula for the Jack polynomials, which form the ...

A (symmetric) nested orthogonal array is a symmetric orthogonal array OA(N, k, s, g) which contains an OA(M,k, r, g) as a subarray, where M < N and r < s. In this communication, some methods of construction of nested symmetric orthogonal arrays are given. Asymmetric nested orthogonal arrays are defined and a few methods of their construction are described.

The search for a canonical set of eigenvectors of the discrete Fourier transform has been ongoing for more than three decades. The goal is to find an orthogonal basis of eigenvectors which would approximate Hermite functions – the eigenfunctions of the continuous Fourier transform. This eigenbasis should also have some degree of analytical tractability and should allow for efficient numerical c...