On the distribution of an arbitrary subset of the eigenvalues for some finite dimensional random matrices

We present some new results on the joint distribution of an arbitrary subset of the ordered eigenvalues of complex Wishart, double Wishart, and Gaussian hermitian random matrices of finite dimensions, using a tensor pseudo-determinant operator. Specifically, we derive compact expressions for the joint probability distribution function of the eigenvalues and the expectation of functions of the eigenvalues, including joint moments, for the case of both ordered and unordered eigenvalues.

EVALUATION OF ELECTROMYOGRAPHIC NOISE STATISTICAL CHARACTERISTICS IN MULTICHANNEL ECG RECORDINGS

Electromyographic noise is one of the most common noises in electrocardiogram. In case of several electrocardiogram leads, electromyographic noise affects each lead to different extent. It can be taken into account when developing algorithms for multilead electrocardiogram record processing. However, in the existing literature, there is no information about the relationship of electromyographic noise in various ECG leads and their joint probability distribution. The purpose of this paper is to study statistical characteristics of electromyographic noise in ECG signal, from which the electromyographic noise is extracted. The paper proposes a method for extracting electromyographic noise from electrocardiogram signal, based on a polynomial approximation of electrocardiogram signal fragments in sliding window with overlapping fragment subsequent weight averaging. Using this method, fragments of electromyographic noise are extracted from multichannel electrocardiogram records. Based on the obtained data, a joint probability distribution function of electromyographic noise in two adjacent leads is selected, and the correlation relationships between the electromyographic noise in various ECG leads are investigated. The results show that the joint probability distribution function of electromyographic noise in two adjacent leads in the first approximation can be described using bivariate normal distribution. In addition, between the samples of electromyographic noise from two adjacent leads quite strong correlation relationships can be observed.

About difference electron densities and their properties

Difference electron densities do not play a central role in modern phase refinement approaches, essentially because of the explosive success of the EDM (electron-density modification) techniques, mainly based on observed electron-density syntheses. Difference densities however have been recently rediscovered in connection with theVLD(Vive la Difference) approach, because they are a strong support for strengthening EDM approaches and forab initiocrystal structure solution. In this paper the properties of the most documented difference electron densities, here denoted asF−Fp,mF−FpandmF−DFpsyntheses, are studied. In addition, a fourth new difference synthesis, here denoted as {\overline F_q} synthesis, is proposed. It comes from the study of the same joint probability distribution function from which theVLDapproach arose. The properties of the {\overline F_q} syntheses are studied and compared with those of the other three syntheses. The results suggest that the {\overline F_q} difference may be a useful tool for making modern phase refinement procedures more efficient.

The method of joint probability distribution functions, neighbourhoods, and representations

Wilson statistics, described in Chapter 2, aims at calculating the distribution of the structure factor P(F) ≡ P(|F|, φ) when nothing is known about the structure; the positivity and atomicity of the electron density (both promoted by the positive nature of the atomic scattering factors fj) are the only necessary assumptions. Wilson results may be synthesized as follows: . . . the modulus R = |E| is distributed according to equations (2.7) or (2.8), while no prevision is possible about φ, which is distributed with constant probability 1/(2π). . . . In other words, knowledge of the R moduli does not provide information about a phase; this agrees well with Section 3.3, according to which experimental data only allow an estimate of s.i. (and also s.s. if the algebraic form of the symmetry operators has been fixed). Let us now consider P(Fh1 , Fh2 ) ≡ P(|Fh1 |, |Fh2 |, φh1 , φh2 ), the joint probability distribution function of two structure factors. If the two structure factors are uncorrelated (i.e. no relation is expected between their moduli and between their phases), P will coincide with the product of two Wilson distributions (2.7) or (2.8), say, . . . P(Fh1 , Fh2 ) ≡ P(|Fh1 |, φh1 ) · P(|Fh2 |, φh2 ) = 1/4π2 P(|Fh1 |)P(|Fh2 |), . . . which is useless (because the two Wilson distributions are useless) for solving the phase problem; indeed, the relation does not provide any phase information. The question is now: if two structure factors are correlated, may their joint probability distribution function be used for solving the phase problem? Let us first use a simple example to show how much additional information (i.e. that is not present in the two elementary distributions) may be stored in a joint probability distribution function; then we will answer the question. Let us suppose that the human population of a village has been submitted to statistical analysis to define how weight and height are distributed.

DSP Acceleration for Dynamic Financial Models

We present an extensive dynamic financial model that encompasses most models used today in finance and economics. We show that this model is a good match to the capabilities of DSP chips. Particularly, DSP is able to perform the high-speed Monte Carlo simulations that are required to solve many large-scale, intractable financial problems. By simulating a sufficiently large number of future scenarios, DSP chips can rapidly achieve a good approximation of the probable future joint probability distribution function of modeled variables. This probability distribution can be used for the valuation of financial derivatives, computing value at risk, studying macroeconomic policy decisions, and many other purposes. DSP enables such simulations to be faster, cooler, greener, and cheaper than ever before.

Multiple-wavelength anomalous dispersion techniques applied to powder data: a probabilistic method for finding the substructureviajoint probability distribution functions

A new joint probability distribution function method is described to find the anomalous scatterer substructure from powder data. The method requires two wavelengths; the conclusive formulas provide estimates of the substructure structure factor moduli, from which the anomalous scatterer positions can be found by Patterson or direct methods. The theory has been preliminarily applied to two compounds, the first having Pt and the second having Fe as anomalous scatterer. Both substructures were correctly identified.