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We study Feynman checkers, the most elementary model of electron motion introduced by  Feynman. For the model, we prove that the probability to find an electron vanishes nowhere inside the light cone. We also prove several results on the average electron velocity. In addition, we present a lot of identities related to the model.

The transport phenomenon (movement and diffusion) of inertia Brownian particles in a periodic potential with non-Gaussian noise is investigated. It is found that proper noise intensity Q will promote particles directional movement (or diffusion), but large Q will inhibit this phenomenon. For large value of Q, the average velocity $〈V〉$ (or the diffusion coefficient D) has a maximum with increasing correlation time $\tau$. But for small value of Q, $〈V〉$ (or D) decreases with increasing $\tau$. In some cases, for the same value of Q and the same value of $\tau$, non-Gaussian noise can induce particles directional movement (or diffusion), but Gaussian colored noise cannot.

The transport phenomenon of coupled self-propelled particles with potential and colored noise is investigated. Large translational motion noise intensity is good for the directional transport in $-x$-direction, but large self-correlation time of translational motion noise will inhibit this transport. For proper value of the asymmetry parameter, coupled inert particles move always in the $-x$-direction with increasing angular noise intensity, but in coupled self-propelled particles appears current reversal phenomenon with increasing angular noise intensity. The average velocity has a maximum and a minimum with increasing spring constant k. For inert particles and particles with small self-propelled speed, large number of particles is good for directional movement, but the effect of coupling will become weak when the self-propelled speed is large.

To study the agitating effect of submersible mixers in a square sewage treatment pool, the three-dimensional modeling Pro/E software was adopted to establish the physical model. The large-scale computational fluid dynamics software FLUENT6.3 was used, and the large-scale software ICEM was selected to build an unstructured tetrahedron grid of the sewage treatment pool. Next, the sewage treatment pool was numerically simulated by RNG k-ε turbulent model and move coordinate system technology. The macrofluid field and the flow field distribution of each section were analyzed to observe the efficiency of each submersible mixer. The average velocity of the fluid and the stirring volume were studied simultaneously. Results show that, under the action of three mixers, fluid of the sewage pool forms a continuous circulating water flow. The fluid is mixed adequately. The average velocity of fluid in the pool is at around 0.3 m/s, and the fluid mixing area in the pool is more than 90%, which is in agreement with the working requirements. Consequently, it can provide a reference basis for the practical engineering application of submersible mixers by using this method.