We consider the boundary Hardy–Hénon equation

\[ -\Delta u=(1-|x|)^{\alpha} u^{p},\ \ x\in B_1(0), \]

$B_1(0)\subset \mathbb {R}^{N}$

$(N\geq 3)$

$1$

$0$

$p>0$

$\alpha \in \mathbb {R}$

$0< p<({N+2})/({N-2})$

$\alpha \leq -2$

$p>1$

$\alpha >-2$

$1< p<({N+2})/({N-2})$

$0< p\leq 1$

$\alpha \leq -2$

$0< p<1$

$\alpha >-2$

Despite a ≥75% reduction in the geographic range of Mongolian gazelles Procapra gutturosa over the past 50 years, the species is still recognized as the most numerous large grassland herbivore in Asia. Its actual population size, however, is still disputed, and we therefore estimated its numbers in an 80,000 km2 area in the eastern steppe of Mongolia by driving long-distance (1,200–1,400 km) transect surveys during spring and autumn 2000–2002. Quantitative estimates of gazelle numbers are essential for understanding the causes of changes in the population, and thus devising conservation strategies to assure its long-term viability. Observed herds ranged in size from 1–4,000; among surveys, median herd sizes varied from 14–42. Density estimates varied from 10.7 gazelles km−2 in spring to 11.5 gazelles km−2 in autumn, with total population estimates of 803,820 (483,790–1,330,100 95% confidence interval) and 870,625 (499,432–1,491,278 95% confidence interval), respectively. Confidence limits were wide, and to obtain a coefficient of variation of 20%, transect lengths would need to be extended three- to four-fold. Until more efficient means for conducting population surveys can be implemented, driving long-distance transects, combined with distance analysis, seem to provide the best quantitative estimate of Mongolian gazelle populations.

In this paper we prove: for any positive integers $a$ and $q$ with $\left( a,\,q \right)\,=\,1$, we have uniformly

$$\sum\limits_{\begin{matrix} n\le N \\ (n,q)=1,n\bar{n}\equiv 1(\,\bmod \,q) \\ \end{matrix}}{\mu (n)e(\frac{a\bar{n}}{q})\ll Nd(q)\left\{ \frac{{{\log }^{\frac{5}{2}}}N}{{{q}^{\frac{1}{2}}}}+\frac{{{q}^{\frac{1}{5}}}{{\log }^{\frac{13}{5}}}N}{{{N}^{\frac{1}{5}}}} \right\}.}$$

This improves the previous bound obtained by D. Hajela, A. Pollington and B. Smith [5].

The importance of reporting estimates and confidence intervals for statistical analyses has been well publicised in the arena of medical studies for some years now. The requirement to give confidence intervals for the main results of a study has been included in the statistical guidelines for contributors to medical journals since the 1980s and methodological points such as this are discussed in the Statistical Notes section of the British Medical Journal. If the use of quantitative methods in British sociology is to be encouraged, as Frank Bechhofer (1996) suggests is needed, it is important to have a forum for the dissemination of basic methodological issues which is accessible to researchers within the discipline. This note aims to achieve such dissemination by using an example from current research to illustrate this fundamental, but often overlooked, aspects of quantitative analysis.

We derive conditions for the time-uniform continuity of random sequences with respect to variations of governing parameters, and also obtain some estimates of the modulus of continuity. We then apply the results to find conditions of continuity for Markov chains with arbitrary state space, and to construct finite approximations to them.

Let Xnn denote the largest of n independent N(0, 1) variables. Several methods of estimating P(Xnn ≦ x) are considered. It is shown that X2nn, when normalized in an optimal way, converges to the extreme value distribution at a rate of only 1/(log n)2, and that if 0 < t ≠ 2 then |Xnn|t converges at a rate of 1/log n. Therefore it is not feasible to use the extreme value distribution to estimate probabilities for normal extremes unless the sample size is extremely large. An alternative approach is presented, which gives very good estimates of P(Xnn ≦ x) for n ≧ 10. The case of rth extremes is also considered.

This paper is concerned with the asymptotic theory of estimates of an unknown parameter in continuous-time Markov processes, which are described by non-linear stochastic differential equations. The maximum likelihood estimate and the minimum contrast estimate are investigated. For these estimates strong consistency and asymptotic normality are proved. The unknown parameter is assumed to take its values either in an open or in a compact set of real numbers.