Convolution operations arising from Vandermonde matrices

Different types of convolution operations involving large Vandermonde matrices are considered. The convolutions parallel those of large Gaussian matrices and additive and multiplicative free convolution. First additive and multiplicative convolution of Vandermonde matrices and deterministic diagonal matrices are considered. After this, several cases of additive and multiplicative convolution of two independent Vandermonde matrices are considered. It is also shown that the convergence of any combination of Vandermonde matrices is almost sure. We will divide the considered convolutions into two types: those which depend on the phase distribution of the Vandermonde matrices, and those which depend only on the spectra of the matrices. A general criterion is presented to find which type applies for any given convolution. A simulation is presented, verifying the results. Implementations of all considered convolutions are provided and discussed, together with the challenges in making these implementations efficient. The implementation is based on the technique of Fourier-Motzkin elimination, and is quite general as it can be applied to virtually any combination of Vandermonde matrices. Generalizations to related random matrices, such as Toeplitz and Hankel matrices, are also discussed.Comment: Submitted to IEEE Transactions on Information Theory. 16 pages, 1 figur

Second OH Overtone Excitation And Statistical Dissociation Dynamics Of Peroxynitrous Acid

The second OH overtone transition of the trans-perp conformer of peroxynitrous acid (tp-HOONO) is identified using infrared action spectroscopy. HOONO is produced by the recombination of photolytically generated OH and NO(2) radicals, and then cooled in a pulsed supersonic expansion. The second overtone transition is assigned to tp-HOONO based on its vibrational frequency (10 195.3 cm(-1)) and rotational band contour, which are in accord with theoretical predictions and previous observations of the first overtone transition. The transition dipole moment associated with the overtone transition is rotated considerably from the OH bond axis, as evident from its hybrid band composition, indicating substantial charge redistribution upon OH stretch excitation. The overtone band exhibits homogeneous line broadening that is attributed to intramolecular vibrational redistribution, arising from the coupling of the initially excited OH stretch to other modes that ultimately lead to dissociation. The quantum state distributions of the OH X (2)Pi (nu=0) products following first and second OH overtone excitation of tp-HOONO are found to be statistical by comparison with three commonly used statistical models. The product state distributions are principally determined by the tp-HOONO binding energy of 16.2(1) kcal mol(-1). Only a small fraction of the OH products are produced in nu=1 following the second overtone excitation, consistent with statistical predictions

The Noncoherent Rician Fading Channel -- Part I : Structure of the Capacity-Achieving Input

Transmission of information over a discrete-time memoryless Rician fading channel is considered where neither the receiver nor the transmitter knows the fading coefficients. First the structure of the capacity-achieving input signals is investigated when the input is constrained to have limited peakedness by imposing either a fourth moment or a peak constraint. When the input is subject to second and fourth moment limitations, it is shown that the capacity-achieving input amplitude distribution is discrete with a finite number of mass points in the low-power regime. A similar discrete structure for the optimal amplitude is proven over the entire SNR range when there is only a peak power constraint. The Rician fading with phase-noise channel model, where there is phase uncertainty in the specular component, is analyzed. For this model it is shown that, with only an average power constraint, the capacity-achieving input amplitude is discrete with a finite number of levels. For the classical average power limited Rician fading channel, it is proven that the optimal input amplitude distribution has bounded support.Comment: To appear in the IEEE Transactions on Wireless Communication

On the accuracy of phase-type approximations of heavy-tailed risk models

Numerical evaluation of ruin probabilities in the classical risk model is an important problem. If claim sizes are heavy-tailed, then such evaluations are challenging. To overcome this, an attractive way is to approximate the claim sizes with a phase-type distribution. What is not clear though is how many phases are enough in order to achieve a specific accuracy in the approximation of the ruin probability. The goals of this paper are to investigate the number of phases required so that we can achieve a pre-specified accuracy for the ruin probability and to provide error bounds. Also, in the special case of a completely monotone claim size distribution we develop an algorithm to estimate the ruin probability by approximating the excess claim size distribution with a hyperexponential one. Finally, we compare our approximation with the heavy traffic and heavy tail approximations.Comment: 24 pages, 13 figures, 8 tables, 38 reference

Detecting and Describing Dynamic Equilibria in Adaptive Networks

We review modeling attempts for the paradigmatic contact process (or SIS model) on adaptive networks. Elaborating on one particular proposed mechanism of topology change (rewiring) and its mean field analysis, we obtain a coarse-grained view of coevolving network topology in the stationary active phase of the system. Introducing an alternative framework applicable to a wide class of adaptive networks, active stationary states are detected, and an extended description of the resulting steady-state statistics is given for three different rewiring schemes. We find that slight modifications of the standard rewiring rule can result in either minuscule or drastic change of steady-state network topologies.Comment: 14 pages, 10 figures; typo in the third of Eqs. (1) correcte