Regressions and monotone chains II: The poset of integer intervals

A regressive function (also called a regression or contractive mapping) on a partial order P is a function mapping P to itself such that (x)x. A monotone k-chain for is a k-chain on which is order-preserving; i.e., a chain x ₁<...ksuch that (x ₁)...(x_k). Let P _nbe the poset of integer intervals {i, i+1, ..., m} contained in {1, 2, ..., n}, ordered by inclusion. Let f(k) be the least value of n such that every regression on P _nhas a monotone k+1-chain, let t(x,j) be defined by t(x, 0)=1 and t(x,j)=x ^t(x,j–1). Then f(k) exists for all k (originally proved by D. White), and t(2,k) < f(K) <t( + _k, k) , where _k 0 as k. Alternatively, the largest k such that every regression on P _nis guaranteed to have a monotone k-chain lies between lg^*(n) and lg^*(n)–2, inclusive, where lg^*(n) is the number of appliations of logarithm base 2 required to reduce n to a negative number. Analogous results hold for choice functions, which are regressions in which every element is mapped to a minimal element.     

Positron studies of hydrogen-defect interactions in proton irradiated molybdenum

Molybdenum single crystals are irradiated at 20 K with 6 MeV protons. The radiation damage and lattice defect annealing is studied by positron lifetime spectroscopy in the temperature range from 15 to 720 K. Loss of vacancies due to recombination with mobile interstitials is observed at 40 K (Stage I) in agreement with resistivity measurements. This is the first time Stage I is observed by positrons below 77 K. The implanted hydrogen decorates the vacancies around 100 K, which is consistent with a hydrogen migration energy in molybdenum:E _M ^H = 0.3–0.4 eV. Clustering of spatially correlated vacancies takes place in a wide temperature region below the usual vacancy clustering stage (Stage III). Stage III is observed at rather low temperatures (400–480 K) due to the very high vacancy concentration. Hydrogen bound to vacancies and vacancy clusters is released above 540 K, which puts an upper limit to the hydrogen binding energy:E _B ^H 1.4 eV. The present work emphasizes the advantage of employing a vacancy sensitive technique to study hydrogen in metals, where its intrinsic solubility is low. In such metals (as molybdenum) both the effective solubility and the effective mobility of hydrogen are strongly influenced by the presence of vacancies.     

Correlations of Active Galactic Nuclei with Microquasars

Correlations of active galactic nuclei (AGNs) with microquasars are discussed based on the coexistence of the Blandford-Znajek (BZ) and magnetic coupling (MC) processes (CEBZMC) in black hole (BH) accretion disk.The proportions of several quantities of BH systems for both AGNs and microquasars are derived by combining the observational data with CEBZMC. It is shown that the square of the magnetic field at the BH horizon is inversely proportional to the BH mass, while the accretion rate of the disk is proportional to the BH mass. In addition, the very steep emissivity indexes from the recent XMM-Newton observations of the nearby bright Seyfert 1 galaxy MCG-6-30-15 and the microquasars XTE J1650-500 are well fitted by considering the MC effects on the disk radiation. These results suggest strongly the correlations of A GNs with microquasars.     

Degradation behavior of electrical properties of GaInAs (1.0 eV) and GaInAs (0.7 eV) sub-cells of IMM4J solar cells under1-MeV electron irradiation

In this work the degradation effects of the Ga_(0.7)In_(0.3)As(1.0 eV) and Ga_(0.42)In_(0.58)As(0.7 eV) sub-cells for IMM4J solar cells are investigated after 1-MeV electron irradiation by using spectral response and photoluminescence(PL) signal amplitude analysis, as well as electrical property measurements. The results show that, compared with the electrical properties of traditional single junction(SJ) GaAs(1.41 eV) solar cell, the electrical properties(such as Isc, Voc, and Pmax)of the newly sub-cells degrade similarly as a function of log ?, where ? represents the electron fluence. It is found that the degradation of Voc is much more than that of Isc in the irradiated Ga_(0.42)In_(0.58)As(0.7 eV) cells due to the additional intrinsic layer, leading to more serious damage to the space charge region. However, of the three types of SJ cells with the gap widths of 0.7, 1.0, and 1.4 eV, the electric properties of the Ga_(0.7)In_(0.3)As(1.0 eV) cell decrease largest under each irradiation fluence. Analysis on the spectral response indicates that the Jsc of the Ga_(0.7)In_(0.3)As(1.0 eV) cell also shows the most severe damage. The PL amplitude measurements qualitatively confirm that the degradation of the effective minority carrier life-time(τeff) in the SJ Ga_(0.7)In_(0.3)As cells is more drastic than that of SJ GaAs cells during the irradiation. Thus,the output current of Ga_(0.7)In_(0.3)As sub-cell should be controlled in the irradiated IMM4J cells.     

Strong approximation of empirical kac processes

Summary A strong approximation result is derived for the Poisson sample size empirical characteristic process. Work done while the author was on leave from Szeged University and a Visiting Scientist at Carleton University, supported by Canadian N.R.C. operating grants of D. A. Dawson and J. N. K. Rao.     

On the joint distribution of two discrete random variables

Summary LetX, Y be two discrete random variables with finite support andX≧Y. Suppose that the conditional distribution ofY givenX can be factorized in a certain way. This paper provides a method of deriving the unique form of the marginal distribution ofX (and hence the joint distribution of (X, Y)) when partial independence only is assumed forY andX−Y.     

Factorizing multivariate time series operators

Motivated by problems occurring in the empirical identification and modelling of a n-dimensional ARMA time series X(t) we study the possibility of obtaining a factorization (I + a₁B + … + a_pB^p) X(t) = [Π_i=1^p (I ? α_iB)] X(t), where B is the backward shift operator. Using a result in [3] we conclude that as in the univariate case such a factorization always exists, but unlike the univariate case in general the factorization is not unique for given a₁, a₂,…, a_p. In fact the number of possibilities is limited upwards by $\text{(np)!}\text{(n!)}{}^{\mathrm{p}}$, there being cases, however, where this maximum is not reached. Implications for the existence and possible use of transformations which removes nonstationarity (or almost nonstationarity) of X(t) are mentioned.     

Markov inequalities on partially ordered spaces

Let X(ω) be a random element taking values in a linear space $\text{X}$ endowed with the partial order ≤; let $\text{G}$₀ be the class of nonnegative order-preserving functions on $\text{X}$ such that, for each g∈$\text{G}$₀, E[g(X)] is defined; and let $\text{G}$₁?$\text{G}$₀ be the subclass of concave functions. A version of Markov's inequality for such spaces in P(X ≥ x) ≤ inf_$\text{G}$0E[g(X)]/g(x). Moreover, if E(X) = ξ is defined and if Jensen's inequality applies, we have a further inequality P(X≥x) ≤ inf_$\text{G}$1E[g(X)]/g(x) ≤ inf_$\text{G}$1g(ξ)/g(x). Applications are given using a variety or orderings of interest in statistics and applied probability.     

A note on realizing homomorphisms of category algebras

The complete Boolean homomorphisms from the category algebra $\text{C}$(X) of a complete matrix space X to the category algebra $\text{C}$(Y) of a Baire topological space Y are characterized as those σ-homomorphisms which are induced by continuous maps from dense G₈-subsets of Y into X. This result is used to deduce a series of related results in topology and measure theory (some of which are well-known). Finally a similar result for the complete Boolean homomorphisms from the category algebra $\text{C}$(X) of a compact Hausdorff space X tothe category algebra $\text{C}$(Y) of a Baire topological space Y is proved.