first passage
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In this paper, we introduce a class of double-weighted polygon networks with two different meanings of weighted factors [Formula: see text] and [Formula: see text], which represent path-difficulty and path-length, respectively, based on actual traffic networks. Picking an arbitrary node from the hub nodes set as the trap node, and the double-weighted polygon networks are divided into [Formula: see text] blocks by combining with the iterative method. According to biased random walks, the calculation expression of average receiving time (ART) of any polygon networks is given by using the intermediate quantity the mean first-passage time (MFPT), which is applicable to any [Formula: see text] ([Formula: see text]) polygon networks. What is more, we display the specific calculation process and results of ART of the double-weighted quadrilateral networks, indicating that ART grows exponentially with respect to the networks order and the exponent is [Formula: see text] which grows with the product of [Formula: see text]. When [Formula: see text] increases, ART increases linearly ([Formula: see text]) or sublinearly ([Formula: see text]) with the size of networks, and the smaller value of [Formula: see text], the higher transportation efficiency.

In order to deepen the understanding of rotating effects on internal cooling, the flow and heat transfer characteristics of 2-pass rotating rectangular smooth/ribbed channels are investigated by Reynolds-Averaged Navier-Stokes (RANS) simulation. Three rotating numbers (Ro = 0.10, 0.25, and 0.40) are simulated, and the maximum buoyancy parameter (Bo) reaches 5.0. The results show that the rotating buoyancy has significant effects on the flow and heat transfer under high Bo conditions. When Bo > 1.0, rotating buoyancy inducts flow separation near the leading edge (LE) in the first passage, while the air flow in the second passage shows a double-peak profile. With increased Bo, the heat transfer in the first passage is greatly increased, and the maximum growth rate occurs at Bo = 0.6~1.0. However, the heat transfer in the second passage has no obvious changes due to a strong turn effect. In the ribbed channel, rotating effects are much weaker than those in the smooth channel. This research helps to improve the understanding of the internal cooling heat transfer mechanism in land-based gas turbines under typical operating conditions.

Abstract Almost all the media the particles move in are non-static, one of which is the most common expanding or contracting (by a scale factor) non-static medium discussed in this paper. Depending on the expected resolution of the studied dynamics and the amplitude of the displacement caused by the non-static media, sometimes the non-static behaviors of the media can not be ignored. In this paper, we build the model describing L\'evy walks in one-dimension uniformly non-static media, where the physical and comoving coordinates are connected by scale factor. We derive the equation governing the probability density function of the position of the particles in comoving coordinate. Using the Hermite orthogonal polynomial expansions, some statistical properties are obtained, such as mean squared displacements (MSDs) in both coordinates and kurtosis. For some representative non-static media and L\'{e}vy walks, the asymptotic behaviors of MSDs in both coordinates are analyzed in detail. The stationary distributions and mean first passage time for some cases are also discussed through numerical simulations.