In this paper, we prove the Strong uniform consistency and asymptotic normality of the kernel density estimator proposed by Jones for length-biased data. The approach is based on the invariance principle for the empirical processes proved by Horvath. All simulations are drawn for different cases to demonstrate both, consistency and asymptotic normality and the method is illustrated by real automobile brake pads data.

‎Let $\{Y_i; i = 1,\ldots,n \}$ be a length-biased sample from a population with cumulative distribution function $F(\cdot)$‎. If the probability of an item selected in the sample is proportional to its length‎, ‎then the distribution of the observed length is known as the length-biased distribution‎.‎We consider the kernel-type estimator $F_n^s(\cdot)$ of $F(\cdot)$‎. ‎Under suitable conditions‎, ‎the extended Glivenko-Cantelli theorem for $F_n^s(\cdot)$ is proved‎.

Kernel density estimators are the basic tools for density estimation in non-parametric statistics. The k-nearest neighbor kernel estimators represent a special form of kernel density estimators, in which the bandwidth is varied depending on the location of the sample points. In this paper, we initially introduce the k-nearest neighbor kernel density estimator in the random left-truncation model, and then prove some of its asymptotic behaviors, such as Strong uniform consistency and asymptotic normality. In particular, we show that the proposed estimator has truncation-free variance. Simulations are presented to illustrate the results and show how the estimator behaves for finite samples. Moreover, the proposed estimator is used to estimate the density function of a real data set.

In this paper introduced the notion of a Strong synchronized system,that is a synchronized system whose there is unique , nite path in Fischer cover labeled synchronizing block. We aim to introduce a class of synchronized systems containing so, cs. Every irreducible so, c shift is an Strong synchronized.

Physicalists and dualists have been unable to give a convincing answer to the mind-body problem, because they sacrifice, respectively, the mental causation and the close relationship between mind and body. Consequently, some recent philosophers, like Timothy O'Connor and Jonathan Jacobs, have turned to the idea of Strong emergentism considereing the mind as an emergent but physical substance that has independent causal powers. If this answer is defensible, it will be a promising approach to solve the mind-body problem. However, there are significant challenges facing the idea of Strong emergentism. In this article, we have two objectives: firstly, to address these challenges and to show that even the most serious one, the Collapse Problem, cannot threaten a particular understanding of Strong emergentism. Secondly, we argue that O'Connor and Jacobs's proposal, although thought-provoking, is not comprehensible and can only be understood clearly when we consider the emergent substance to be not merely physical.

This paper describes current activities in the field of Strong motion research in Iran. The seismotectonics and seismological background is explained and current Strong motion monitoring in Iran is summarised. The Strong Ground Motion Network of Iran started its activities since 1973, and has recorded more than 4200 accelerograms. The highest ground acceleration recorded in this network was 1g at Zanjiran station in 1994 Zanjiran earthquake. At presentthe network consists of 986 digital and 71 analogue accelerograph. ISMN has been installed 92 accelerographs in 54 dams. The geophysical and geotechnical investigations for site geology have been conducted in 100 accelerograph stations.