Abstract-A half-range orthogonality theorem concerning the established elementary solutions of the two-group neutron transport equation for isotropic scattering and plane geometry is proved. The orthogonality relations are based on a Chandrasekhar-type H-matrix, and all necessary normalization integrals are evaluated so that the desired expansion coefficients may be expressed concisely in terms...

We introduce non-negative matrix factorization with orthogonality constraints (NMFOC) for detection of a target spectrum in a given set of Raman spectra data. An orthogonality measure is defined and two different orthogonality constraints are imposed on the standard NMF to incorporate prior information into the estimation and hence to facilitate the subsequent detection procedure. Both multipli...

Abstract. The five parameter family of multivariable Askey-Wilson polynomials is studied with four parameters generically complex. The multivariable Askey-Wilson polynomials form an orthogonal system with respect to an explicit (in general complex) measure. A partially discrete orthogonality measure is obtained by shifting the contour to the torus while picking up residues. A parameter domain i...

Orthogonality plays a fundamental role in various mathematical theorems and in physics. The orthogonal eigenfunctions that represent the intrinsic motions of various physical systems can also be regarded as transverse wave modes in a straight waveguide. Because of their orthogonality, these modes propagate independently, without mutual interference. When the wall separation fluctuates, the form...

Multidimensional model describing the " cosmological " and/ or spherically symmetric configuration with (n + 1) Einstein spaces in the theory with several scalar fields and forms is considered. When electromagnetic composite p-brane ansatz is adopted, n " internal " spaces are Ricci-flat, one space M 0 has a non-zero curvature, and all p-branes do not " live " in M 0 , a class of exact solution...

Multidimensional model describing the cosmological evolution and/ or spherically symmetric configuration with (n+1) Einstein spaces in the theory with several scalar fields and forms is considered. When electro-magnetic composite p-brane ansatz is adopted, n ”internal” spaces are Ricci-flat, one space M0 has a non-zero curvature, and all p-branes do not ”live” in M0, a class of exact solutions ...

<p style='text-indent:20px;'>We study the deformation of input space by a trained autoencoder via Jacobians weight matrices. In doing so, we prove bounds for mean squared errors points in space, under assumptions regarding orthogonality eigenvectors. We also show that trace and product eigenvalues Jacobian matrices is good predictor on test points. This dataset independent means testing a...