We consider the hydrodynamic scaling behavior of the mass density with respect to a general class of mass conservative interacting particle systems on ${\mathbb Z}^n$, where the jump rates are asymmetric and long-range of order $\|x\|^{-(n+\alpha)}$ for a particle displacement of order $\|x\|$. Two types of evolution equations are identified depending on the strength of the long-range asymmetry. When $0<\alpha<1$, we find a new integro-partial differential hydrodynamic equation, in an anomalous space-time scale. On the other hand, when $\alpha\geq 1$, we derive a Burgers hydrodynamic equation, as in the finite-range setting, in Euler scale.
Whereas the atmospheric diffusion of particulates is Lagrangian in nature and Lagrangian measurements of the turbulence associated with their dispersion are difficult to obtain, fixed point Eulerian measurements of turbulence are relatively simpler to perform. Results are reported from Eulerian and Lagrangian observations of over-water diffusion experiments conducted off the south shore of Long Island. Oil fog smoke was used as a tracer released at a height of about 6 m from a boat anchored offshore. Concentration measurements were made with another boat or a vehicle on the beach using optical densitometers, similar in operation to integrating nephlometers. Statistical parameters of the meandering plume were obtained. One of the parameters estimated from the concentration measurements across the plume was the lateral standard deviation. The travel distance from the source to the line of measurements varied from 0.5 to 6 km. Meteorological measurements were made on the beach at heights of 16 or 24 m.
This paper discusses the role of the atmospheric boundary layer in the Bay of Bengal cyclogenesis processes. The boundary layer experiment at Digha consisted of three main components: (1) a 10 m micrometeorological tower at the beach with instruments to observe turbulent fluxes of heat and momentum over the ocean; (2) a weather station that continuously recorded mean parameters; and (3) pilot balloon observations to a height of about 1000 m.
A sequence of independent random variables $\{X_1,X_2,\ldots\}$ is called a $B-$harmonic Bernoulli sequence if $P(X_i=1)=1-P(X_i=0) = 1/(i+B)\ i=1,2,\ldots$, with $B\ge 0$. For $k\ge 1$, the count variable $Z_k$ is the number of occurrences of the $k$-string $(1,\protect\underbrace{0,\ldots,0}_{k-1},1)$\vadjust{\vspace*{-2pt}} in the Bernoulli sequence\ldots\$. This paper gives the joint distribution $P_B$ of the count vector ${\bf Z} = (Z_1,Z_2,\ldots)$ of strings of all lengths in a $B-$harmonic Bernoulli sequence. This distribution can be described as follows. There is random variable $V$ with a Beta$(B,1)$ distribution, and given $V=v$, the conditional distribution of ${\bf Z}$ is that of independent Poissons with intensities $(1 -v),\ (1 - v^2)/2,\ (1-v^3)/3, \ldots$. Around 1996, Persi Diaconis stated and proved that when $B=0$, the distribution of $Z_1$ is Poisson with intensity $1$. Emery gave an alternative proof a few months later. For the case $B=0$, it was also recognized that $Z_1,Z_2,\ldots,Z_n$ are independent Poissons with intensities $1, \frac{1}{2},\ldots, \frac{1}{n}$. Proofs up until this time made use of hard combinational techniques. A few years later, Joffe et al, obtained the marginal distribution of $Z_1$ as a Beta-Poisson mixture when $B\geq 0$. Their proof recognizes an underlying inhomogeneous Markov chain and uses moment generating functions. In this note, we give a compact expression for the joint factorial moment of\break $(Z_1,\ldots,Z_N)$ which leads to the joint distribution given above. One might feel that if $Z_1$ is large, it will exhaust the number of $1$'s in the Bernoulli sequence $(X_1,X_2,\ldots)$ and this in turn would favor smaller values for $Z_2$ and introduce some negative dependence. We show that, on the contrary, the joint distribution of ${\bf Z}$ is positively associated or possesses the FKG property.
The design and construction of a bivane to measure horizontal and elevation angle fluctuations in the atmosphere are presented. Wind tunnel tests indicated a reasonably high-frequency response. Field comparisons with a commercially available instrument gave good results. A unique feature in the design of this bivane is its ruggedness in combination with good frequency response.
Consider a tagged particle in zero-range dynamics on the integer lattice in dimension d with rate g whose finite-range jump probabilities p possess a drift. We show, in equilibrium, that the variance of the tagged particle position at time t is at least order t in all dimensions and at most order t in d=1 and d larger or equal to 3 for a wide class of rates g. Also, in d=1, when the jump distribution p is totally asymmetric and nearest-neighbor, and when the rate g(k) increases and g(k)/k decreases with k, we show the diffusively scaled centered tagged particle position converges to a Brownian motion.
In a general stochastic multistate promoter model of dynamic mRNA/protein interactions, we identify the stationary joint distribution of the promoter state, mRNA, and protein levels through an explicit `stick-breaking' construction of interest in itself. This derivation is a constructive advance over previous work where the stationary distribution is solved only in restricted cases. Moreover, the stick-breaking construction allows to sample directly from the stationary distribution, permitting inference procedures and model selection. In this context, we discuss numerical Bayesian experiments to illustrate the results.
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