We answer in a probabilistic setting two questions raised by Stokolos in a private communication. Precisely, given a sequence of random variables $\left\{X_k : k \geq 1\right\}$ uniformly distributed in $(0,1)$ and independent, we consider the following random sets of directions

\begin{equation*}\Omega_{\text{rand},\text{lin}} := \left\{ \frac{\pi X_k}{k}: k \geq 1\right\}\end{equation*}

\begin{equation*}\Omega_{\text{rand},\text{lac}} := \left\{\frac{\pi X_k}{2^k} : k\geq 1 \right\}.\end{equation*}

We prove that almost surely the directional maximal operators associated to those sets of directions are not bounded on $L^p({\mathbb{R}}^2)$ for any $1 \lt p \lt \infty$.

Seasonal variation of temperature, pressure, snow accumulation, winds, and their harmonic analysis are presented by using the data from Zhongshan Station and three Automatic Weather Stations deployed between the East Antarctic coast and the summit of the ice sheet at Dome A for the period 2005–07. Results show that: 1) temperature, snow accumulation and specific humidity decrease with increasing elevation and distance from the coast, with snow accumulation decreasing from 199 mm water equivalent (w.e.) yr-1 at LGB69 (180 km from the coast) to 31 mm w.e. yr-1 at Dome A, 2) Dome A experiences an extremely low minimum temperature of -82.5°C with the monthly mean temperature below -50°C for eight months in contrast to Zhongshan Station which does not show any monthly mean temperatures below -20°C, 3) mean surface wind speed increases from the coast to the escarpment region, and then reduces rapidly towards the interior plateau with the strongest winds occurring at katabatic sites with the greatest surface slopes, 4) temperature and pressure all shows a distinct biannual oscillation with a main minimum in spring and a secondary minimum in autumn, differing slightly from station to station, and 5) winter temperature corelessness increases as a function of elevation and distance from the coast, from 0.260 at the coastal Zhongshan Station to 0.433 at Dome A.

Ganglion cell activity in response to drifting or alternating stimulus patterns was recorded in an eyecup preparation of the turtle. The stimuli were sinusoidal gratings with various spatial frequencies and spatial phases. Spike histograms of the activity were Fourier-analyzed. The results indicated that the retinal circuitry always functioned in a nonlinear manner. The average, first, and second harmonics of the activity were all a function of stimulus conditions, including spatial phase. Similar outcomes were obtained with full-field (retina-wide) and a small-field (1 mm diameter) stimulation. In the middle of the spatial-frequency range, the ratio of second to first harmonic activity was higher in response to static, contrast-reversing gratings than for drifting, constant-contrast ones. The findings suggest that, as distinct from the usual X, Y, and W segregation of ganglion cell properties in other retinas, there is a single class of cells whose responses seem to have attributes of one or another of the usual categories depending on stimulus characteristics. This may be the result of a degree of stimulus-dependent functional modification in the retinal input circuitry to the turtle ganglion cells.

This paper studies the Laplacian operator on a metrized graph, and its spectral theory.

Let $X=G/H$ be a reductive symmetric space and $K$ a maximal compact subgroup of $G$. We study Fourier transforms of compactly supported $K$-finite distributions on $X$ and characterize the image of the space of such distributions.

Consider N towers each made up of a number of counters. At each step a tower is chosen at random, a counter removed which is then added to another tower also chosen at random. The probability distribution for the time needed to empty one of the towers is obtained in the case N = 3. Arguments are set forward as to why no simple formulae can be expected for N > 3. An asymptotic expression for the mean time before one of the towers becomes empty is derived in the case of four towers when they all initially contain a comparably large number of counters. We then study related problems, in particular the ruin problem for three players. Here we use simple martingale methodology as well as a solution proposed by T. S. Ferguson for a slightly modified problem. Throughout the paper it is our main objective to shed light on the reasons why the case N > 3 is so substantially different from the case N ≤ 3.

Let K be a finite graph with vertex set V = {x0, x1, …, xσ–1} and automorphism group G. It is assumed that G acts transitively on V. We can imagine that the vertices of K represent σ cities and a traveler visits the cities in a series of random flights. The traveler starts at a given city and in each flight, independently of the past journey, chooses a city at random as the destination. Denote by vn (n = 1, 2, …) the location of the traveler at the end of the nth flight, and by v0 the initial location. It is assumed that the transition probabilities P{vn = xj | vn–1 = xi}, xi ϵ V, xj ϵ V, do not depend on n and are invariant under the action of G on V. The main result of this paper consists in determining p(n), the probability that the traveler returns to the initial position at the end of the nth flight.