We present a continuous time generalization of a random walk with complete memory of its

We obtain compact, exact, analytical expressions for the first-passage-time distribution for a particle diffusing on a planar wedge for special values of the wedge angle. Specifically, we calculate the first-passage-time distribution for the diffusing particle through a planar wedge of angle pi/n, where n is an integer. For the cases n=2 and n odd, we provide an exact closed-form expression to the first-passage-time distribution while for the remaining cases, we provide it in integral form and evaluate numerically using quadratures. We then show that our results are in good agreement with Markovian simulations in the continuum limit.

We give an exact solution to the generalized Langevin equation of motion of a charged Brownian particle in a uniform magnetic field that is driven internally by an exponentially correlated stochastic force. A strong dissipation regime is described in which the ensemble-averaged fluctuations of the velocity exhibit transient oscillations that arise from memory effects. Also, we calculate generalized diffusion coefficients describing the transport of these particles and briefly discuss how they are affected by the magnetic field strength and correlation time. Our asymptotic results are extended to the general case of internal driving by correlated Gaussian stochastic forces with finite autocorrelation times.

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We solved the wind-influenced projectile motion problem with the same initial and final heights and obtained exact analytical expressions for the shape of the trajectory, range, maximum height, time of flight, time of ascent, and time of descent with the help of the Lambert W function. It turns out that the range and maximum horizontal displacement are not always equal. When launched at a critical angle, the projectile will return to its starting position. It turns out that a launch angle of 90°maximizes the time of flight, time of ascent, time of descent, and maximum height and that the launch angle corresponding to maximum range can be obtained by solving a transcendental equation. Finally, we expressed in a parametric equation the locus of points corresponding to maximum heights for projectiles launched from the ground with the same initial speed in all directions. We used the results to estimate how much a moderate wind can modify a golf ball's range and suggested other possible applications.

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We study minimal mean-field models of viral drug resistance development in which the efficacy of a therapy is described by a one-dimensional stochastic resetting process with mixed reflecting-absorbing boundary conditions. We derive analytical expressions for the mean survival time for the virus to develop complete resistance to the drug. We show that the optimal therapy resetting rates that achieve a minimum and maximum mean survival times undergo a second- and first-order phase transition-like behaviour as a function of the therapy efficacy drift. We illustrate our results with simulations of a population dynamics model of HIV-1 infection.

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