Orthogonal Bases and the QRAlgorithm

Construction of Compactly Supported Nonseparable Orthogonal Wavelets of L^2(R^n)

Analysis of High-order Approximations by Spectral Interpolation Applied to One- and Two-dimensional Finite Element Method

The implementation of high-order (spectral) approximations associated with FEM is an approach to overcome the difficulties encountered in the numerical analysis of complex problems. This paper proposes the use of the spectral finite element method, originally developed for computational fluid dynamics problems, to achieve improved solutions for these types of problems. Here, the interpolation n...

New construction of mutually unbiased bases in square dimensions

We show that k = w + 2 mutually unbiased bases can be constructed in any square dimension d = s provided that there are w mutually orthogonal Latin squares of order s. The construction combines the design-theoretic objects (k, s)-nets (which can be constructed from w mutually orthogonal Latin squares of order s and vice versa) and generalized Hadamard matrices of size s. Using known lower bound...

Constructing Mutually Unbiased Bases from Quantum Latin Squares

We introduce orthogonal quantum Latin squares, which restrict to traditional orthogonal Latin squares, and investigate their application in quantum information science. We use quantum Latin squares to build maximally entangled bases, and show how mutually unbiased maximally entangled bases can be constructed in square dimension from orthogonal quantum Latin squares. We also compare our construc...

Pseudo Orthogonal Bases Give the Optimal Solution to Active Learning in Neural Networks

Pseudo orthogonal bases are a certain type of frames proposed in the engineering field, whose concept is equivalent to a normalized tight frame in the frame terminology. This paper shows that pseudo orthogonal bases play an essential role in neural network learning. One of the most important issues in neural network learning is “what training data provides the optimal generalization capability?...