Orthogonal Transformation

An orthonormal transformation in N-dimensional space can be obtained by choosing a set of N mutually orthogonal vectors and then normalizing their lengths to make them unit vectors. Alternatively, we can start from N orthogonal continuous functions, unifromly sample each of them appropriately ensuring that the N-samples of each of them treated as vectors are still orthogonal to each other and then normalizing those vectors to form the basis. Orthogonality of sinusoids with rational frequency ratio: With continuous sinusoidal functions, the area under the curve obtained by the product of two sinusoids of frequency equal to different integer multiples of the same fundamental frequency is zero over one period of the fundamental frequency.

A model for the adsorption of phenol in a fixed bed of a polymeric adsorbent is developed. Model parameters (equilibrium parameters, capacity factor, axial dispersion, film mass transfer coefficient, and intraparticle effective diffusivity) are experimentally determined from independent experiments. Numerical solution of the model equations uses the method of lines with double orthogonal collocation in finite elements. The model is used for the prediction of breakthrough curves and is part of a package for the design of cyclic processes.

Constant modulus transforms like discrete Fourier transform (DFT), Walsh transform, and Gold codes have been successfully used over several decades in several engineering applications, including discrete multi-tone (DMT), orthogonal frequency division multiplexing (OFDM) and code division multiple access (CDMA) communications systems. In this paper, we present a generalized framework for DFT called Generalized DFT (GDFT) with nonlinear phase by exploiting the phase space. We show that GDFT offers sizable correlation improvements over DFT, Walsh, Oppermann and Gold codes, leading to better performance in all multi-carrier communications scenarios investigated. We also highlight how known constant modulus orthogonal transforms are special solutions of the proposed GDFT framework. Moreover, we introduce practical design methods offering computationally efficient implementations of GDFT as enhancements to DFT. We conclude the paper with examples of communications applications where GDFT is shown to outperform DFT and other known constant modulus bases.

The problem of modeling complex processes with a large number of inputs is addressed. A new method is proposed for the optimization of the models in minimum Cp statistic sense using QR with a modified scheme of column pivoting (m-QRcp) factorization. Two different classes of multilayer nonlinear modeling problems are explored: 1) in the first class of models, each layer comprises multiple linearly parameterized submodels or cells; the individual cells are optimally modeled using QR factorization, and m-QRcp factorization ensures optimal selection of variables across the layers. 2) The nonhomogeneous feed-forward neural network is chosen as the second class of models, where the network architecture and structure are optimized in terms of best set of hidden links (and nodes) using m-QPcp factorization. In both the cases, the optimization is shown to be direct and conclusive. The proposed is a generic approach to the optimal modeling of complex multilayered architectures, which leads to computationally fast and numerically robust parsimonious designs, free from collinearity problems. The method is largely free from heuristics and is amenable to automated modeling

This paper shows how to obtain the spinor field and dynamics from the vielbein and geometry of General Relativity. The spinor field is physically realized as an orthogonal transformation of the vielbein, and the spinor action enters as the requirement that the unit time form be the gradient of a scalar time field.