Narrow Results

Let $f:{𝕊}^1\to {𝕊}^1$ be a $C^{2+𝜖}$ expanding map of the circle and let $v:{𝕊}^1\to ℝ$ be a $C^{1+𝜖}$ function. Consider the twisted cohomological equation $v=\alpha \circ f-Df\cdot \alpha ,$ which has a unique bounded solution $\alpha$. We show that $\alpha$ is either $C^{1+𝜖}$ or continuous but nowhere differentiable. If $\alpha$ is nowhere differentiable then the Newton quotients of $\alpha$, after an appropriated normalization, converges in distribution (with respect to the unique absolutely continuous invariant probability of $f$) to the normal distribution. In particular, $\alpha$ is not a Lipschitz continuous function on any subset with positive Lebesgue measure.

For a rotation by an irrational $\alpha$ on the circle and a BV function $\phi$, we study the variance of the ergodic sums $S_L\phi (x):={\sum }_{j=0}^{L-1}\phi (x+j\alpha )$. When $\alpha$ is not of constant type, we construct sequences $(L_N)$ such that, at some scale, the ergodic sums $S_{L_N}\phi$ satisfy an ASIP. Explicit non-degenerate examples are given with an application to the rectangular periodic billiard in the plane.

The major focus of this study is to describe the structure of a solution designed for robustly detecting and delineating the arterial blood pressure (ABP) signal events. To meet this end, first, the original ABP signal is pre-processed by application of à trous discrete wavelet transform (DWT) for extracting several dyadic scales. Then, a fixed sample size sliding window is moved on the appropriately selected scale and in each slid, six features namely as summation of the nonlinearly amplified Hilbert transform, summation of absolute first-order differentiation, summation of absolute second-order differentiation, curve length, area and variance of the excerpted segment are calculated. Then, all feature trends are normalized and utilized to construct a newly proposed principal components analyzed geometric index (PCAGI) (to be used as the segmentation decision statistic (DS)) by application of a linear orthonormal projection. After application of an adaptive-nonlinear transformation for making the DS baseline stationary, the histogram parameters of the enhanced DS are used to regulate the α-level Neyman–Pearson classifier for false alarm probability (FAP)-bounded delineation of the ABP events. In order to illustrate the capabilities of the presented algorithm, it was applied to all 18 subjects of the MIT-BIH Polysomnographic Database (359,000 beats) and the end-systolic and end-diastolic locations of the ABP signal as well as dicrotic notch pressure were extracted and values of sensitivity and positive predictivity Se = 99.86% and P+ = 99.95% were obtained for the detection of all ABP events. High robustness against measurement noises, acceptable detection-delineation accuracy of the ABP events in the presence of severe heart valvular and arrhythmic dysfunctions within a tolerable computational burden (processing time) and having no parameters dependency to the acquisition sampling frequency can be mentioned as the important merits and capabilities of the proposed PCAGI-based ABP events detection-segmentation algorithm.

In this note, we evaluate the variance of the distribution of Heegner points asymptotically. More precisely, we show

Two fields of research have found tremendous applicability in the analysis of biological data-statistics and information theory. Statistics is extensively used for the measurement of central tendency, dispersion, comparison and covariation. Measures of information are used to study diversity and equitability. These two fields have been used independent of each other for data analysis. In this communication, we develop the link between the two and prove that statistical measures can be used as information measures. Our study will be a new interdisciplinary field of research and it will be possible to describe information content of a system from its statistics.

We determine the covariance of the weight distribution in level 1 Demazure modules of the affine Kac–Moody algebra . This allows us to prove a weak law of large numbers for these weight distributions, and leads to a conjecture about the asymptotic concentration of weights for arbitrary Demazure modules.

We consider families of life distributions with the first three moments belonging to small neighborhoods of respective powers of a positive number. For various shapes of the neighborhoods, we determine exact convergence rates of their Prokhorov radii to zero. This provides a refined evaluation of the effect of specific moment convergence on the weak one.

We study the joint linear complexity of linear recurring multisequences, i.e., of multisequences consisting of linear recurring sequences. The expectation and variance of the joint linear complexity of random linear recurring multisequences are determined. These results extend the corresponding results on the expectation and variance of the joint linear complexity of random periodic multisequences. Then we enumerate the linear recurring multisequences with fixed joint linear complexity and determine the generating polynomial for the distribution of joint linear complexities. The proofs use new methods that enable us to obtain results of great generality.

An ODE model of HTLV-I infection with CD4⁺ T cells is considered and analyzed. The model system has three equilibria: infection free equilibrium, uninfected equilibrium with leukemia cell only and infected equilibrium. It is observed that the infection persists (infected equilibrium is g.a.s.) if basic reproduction number is bigger than one. This model is further modified by introducing discrete time delay to account for the latent period of infection and it is shown that this delay has no destabilizing effect on the local stability of infected equilibrium. Again, this delay model system is extended to corresponding stochastic model considering the randomness in proliferation of leukemia cells. It is done by introducing white noise term in proliferation rate of Leukemia cells and corresponding stochastic delay differential equation model is analyzed using Fourier Transform technique. We observe fluctuations only in Leukemia cells populations and corresponding variations are obtained analytically. A numerical experimentation is performed to analyse and explain the outcomes. We observe that the fluctuations in Leukemia cells’ population are contributed by white noise perturbation of proliferation rate and this has no impact of both uninfected and infected CD4⁺ T cells.

In this and next two chapters, we will use advanced calculus to formalize and extend the probability theory introduced in Chapter 2. The use of calculus enables us to investigate probability more deeply. A number of quantitative-oriented probability concepts will be introduced. In this chapter, we first introduce the concept of a random variable and characterize the probability distributions of a random variable and functions of a random variable by the cumulative distribution function, the probability mass function or probability density function, the moment generating function and the characteristic function, respectively. We also introduce a class of moments and discuss their relationships with a probability distribution. This chapter focuses on univariate distributions.

This paper proposes an unconventional method (PMM) for evaluating the uncertainty of the estimator of measurand value obtained from the non-Gaussian distributed samples of measurement data with a priori partial description (unknown PDF). This method of statistical estimation is based on the apparatus of stochastic polynomial maximization and uses the higher-order statistics (moment and cumulant description) of random variables. The analytical expressions for estimates of uncertainty, obtained with use the polynomial of the degree r = 2 for samples from population of asymmetrical pdf and degree r = 3 — for symmetrical pdf, are given. It is shown that these uncertainties are generally smaller than the uncertainty based only on the arithmetic average, as it is in GUM. Reducing the value of estimated uncertainty of measurement depends on the skewness and kurtosis of samples from asymmetrical pdf or on kurtosis and six order moment of samples from symmetrical pdf. The results of statistical modeling carried out on the basis of the Monte Carlo method confirm the effectiveness of the proposed approach.

This paper studies variance estimators of cross-validation estimators of the generalization error. Three estimators are discussed, and their performance is evaluated in a variety of data models and data sizes. It is shown that the standard error associated with the moment approximation estimator is smaller than that associated with the other two. The effect of training and test set size on these estimators is discussed.