Arbitrary Positive Definite Matrix Research Articles

Error Exponents for Dimension-Matched Vector Multiple Access Channels With Additive Noise

We analyze a class of vector multiple access channels with additive noise, where the sum of the dimensions of the transmitted signals matches that of the received signal. We first focus on the case without power constraints, in the Poltyrev sense, using point process techniques. We find the Poltyrev capacity region for noise processes that are independent and identically distributed over channel uses. For each rate vector strictly in the Poltyrev capacity region, we study, for each subset of the transmitters, the exponent of the decay in block length of the smallest possible probability that decoding results in error for each transmitter in that subset. In the case of independent and identically distributed Gaussian noise, with arbitrary positive definite covariance matrix, we derive random coding exponents for each type of error event—these are lower bounds to the true error exponents. This also leads to random coding error exponents in the traditional power-constrained case, where the power constraint at each transmitter is defined by an arbitrary positive definite matrix.

A generalization of the löwdin orthogonalization

The Lowdin orthonormalized set consists of those unique orthonormal functions which minimize the quantity Σi∫|gi — fi|2dv, i.e., the sum of the squared distances in the Hilbert space between each initial function ff and a corresponding function gg of the orthonormal set. In this paper, the Lowdin orthogonalization is extended to a more general transformation. The new transformation also minimizes the sum Σi∫|gi — fi|2dvdv, but the overlap matrix of the resultant functions gg is an arbitrary positive definite matrix in contrast to the unit matrix in the Lowdin orthogonalization. The new orbital set may be useful as a basis set for some many-electron problems in which the smallness of the sum Σi∫|gi — fi|2dv and a particular form of the overlap matrix are requested at the same time.

Influential observation in the growth curve model with unstructured covariance matrix

Under a normal assumption, Liski (Biometrics 47 (1991) 659–668) gave some measurements for assessing influential observations in the growth curve model (GCM) with a known covariance. For the GCM with an unstructured covariance matrix (UCM), i.e., the covariance is an arbitrary positive definite matrix, the problems of detecting influential observations are discussed in this paper. Based on the empirical influence function of the regression coefficient, a generalized Cook's distance is proposed to measure the influence of a subset of observations on the MLE's fit. In order to make comparison with the generalized Cook's distance, some other diagnostic measurements for assessing the influence are investigated too, which are based on the change of the confidence ellipsoid's volume after deleting the observation's subset. In addition, the influences on some linear combinations of the regression coefficient are discussed. For illustration purpose, a numerical example is provided and the analysis results show that the approaches presented in this paper are useful in practice.

Eigenvalue placement in a specified sector for variable structure control systems

The design of the sliding hyperplanes of a variable structure control system is an important part of the controller design process, and will be considered in this paper. The method outlined here results in the eigenvalues of the closed-loop system during sliding lying in a specified sector in the left-hand half-plane. Numerical examples are given, and the effect of an arbitrary positive definite matrix in a complex Riccati equation on the positioning of the eigenvalues within the sector, is investigated.