Abstract:
In this paper, we compute the non-detection probability of a randomly moving target by a stationary or moving searcher in a square search region. We find that when the searcher is stationary, the decay rate of the non-detection probability achieves the maximum value when the searcher is fixed at the center of the square search region; when both the searcher and the target diffuse with significant diffusion coefficients, the decay rate of the non-detection probability only depends on the sum of the diffusion coefficients of the target and searcher. When the searcher moves along prescribed deterministic tracks, our study shows that the fastest decay of the non-detection probability is achieved when the searcher scans horizontally and vertically.

Abstract:
Current control methods for nonlinear stochastic system have shortcomings such as complex design procedures, high costs, and lack of stability and convergence proof. Considering these problems, based on the equivalent nonlinear system method for obtaining the approximate stationary solutions of the nonlinear stochastic systems, this paper proposes an innovative design procedure for the feedback control of stochastic nonlinear system. The control aims at making the statistical information of the output steady-state probability density function (SPDF) follow those of a target SPDF. Lyapunov function method is presented to prove that the controlled systems do approach to the pre-specified SPDF. Simulation results are given to illustrate the procedure and demonstrate the efficiency.

Abstract:
A single target is hidden at a location chosen from a predetermined probability distribution. Then, a searcher must find a second probability distribution from which random search points are sampled such that the target is found in the minimum number of trials. Here it will be shown that if the searcher must get very close to the target to find it, then the best search distribution is proportional to the square root of the target distribution. For a Gaussian target distribution, the optimum search distribution is approximately a Gaussian with a standard deviation that varies inversely with how close the searcher must be to the target to find it. For a network, where the searcher randomly samples nodes and looks for the fixed target along edges, the optimum is to either sample a node with probability proportional to the square root of the out degree plus one or not at all.

Abstract:
Let a lattice gas of constant density, described by the symmetric simple exclusion process, be brought in contact with a "target": a spherical absorber of radius $R$. Employing the macroscopic fluctuation theory (MFT), we evaluate the probability ${\mathcal P}(T)$ that no gas particle hits the target until a long but finite time $T$. We also find the most likely gas density history conditional on the non-hitting. The results depend on the dimension of space $d$ and on the rescaled parameter $\ell=R/\sqrt{D_0T}$, where $D_0$ is the gas diffusivity. For small $\ell$ and $d>2$, ${\mathcal P}(T)$ is determined by an exact stationary solution of the MFT equations that we find. For large $\ell$, and for any $\ell$ in one dimension, the relevant MFT solutions are non-stationary. In this case $\ln {\mathcal P}(T)$ scales differently with relevant parameters, and it also depends on whether the initial condition is random or deterministic. The latter effects also occur if the lattice gas is composed of non-interacting random walkers. Finally, we extend the formalism to a whole class of diffusive gases of interacting particles.

Abstract:
The survival probability of immobile targets, annihilated by a population of random walkers on inhomogeneous discrete structures, such as disordered solids, glasses, fractals, polymer networks and gels, is analytically investigated. It is shown that, while it cannot in general be related to the number of distinct visited points, as in the case of homogeneous lattices, in the case of bounded coordination numbers its asymptotic behaviour at large times can still be expressed in terms of the spectral dimension $\widetilde {d}$, and its exact analytical expression is given. The results show that the asymptotic survival probability is site independent on recurrent structures ($\widetilde{d}\leq2$), while on transient structures ($\widetilde{d}>2$) it can strongly depend on the target position, and such a dependence is explicitly calculated.

Abstract:
We present a dynamic programming-based solution to the problem of maximizing the probability of attaining a target set before hitting a cemetery set for a discrete-time Markov control process. Under mild hypotheses we establish that there exists a deterministic stationary policy that achieves the maximum value of this probability. We demonstrate how the maximization of this probability can be computed through the maximization of an expected total reward until the first hitting time to either the target or the cemetery set. Martingale characterizations of thrifty, equalizing, and optimal policies in the context of our problem are also established.

Abstract:
Kolmogorov complexity and algorithmic probability are defined only up to an additive resp. multiplicative constant, since their actual values depend on the choice of the universal reference computer. In this paper, we analyze a natural approach to eliminate this machine-dependence. Our method is to assign algorithmic probabilities to the different computers themselves, based on the idea that "unnatural" computers should be hard to emulate. Therefore, we study the Markov process of universal computers randomly emulating each other. The corresponding stationary distribution, if it existed, would give a natural and machine-independent probability measure on the computers, and also on the binary strings. Unfortunately, we show that no stationary distribution exists on the set of all computers; thus, this method cannot eliminate machine-dependence. Moreover, we show that the reason for failure has a clear and interesting physical interpretation, suggesting that every other conceivable attempt to get rid of those additive constants must fail in principle, too. However, we show that restricting to some subclass of computers might help to get rid of some amount of machine-dependence in some situations, and the resulting stationary computer and string probabilities have beautiful properties.

Abstract:
We calculate the survival probability P_S(t) up to time t of a tracer particle moving along a deterministic trajectory in a continuous d-dimensional space in the presence of diffusing but mutually noninteracting traps. In particular, for a tracer particle moving ballistically with a constant velocity c, we obtain an exact expression for P_S(t), valid for all t, for d<2. For d \geq 2, we obtain the leading asymptotic behavior of P_S(t) for large t. In all cases, P_S(t) decays exponentially for large t, P_S(t) \sim \exp(-\theta t). We provide an explicit exact expression for the exponent \theta in dimensions d \leq 2, and for the physically relevant case, d=3, as a function of the system parameters.

Abstract:
Cryptography and simulation of systems require that events of pre-defined probability be generated. This paper presents methods to generate target probability events based on the oblivious transfer protocol and target probabilistic sequences using probability distribution functions.

Abstract:
In this work, we study the target detection and tracking problem in mobile sensor networks, where the performance metrics of interest are probability of detection and tracking coverage, when the target can be stationary or mobile and its duration is finite. We propose a physical coverage-based mobility model, where the mobile sensor nodes move such that the overlap between the covered areas by different mobile nodes is small. It is shown that for stationary target scenario the proposed mobility model can achieve a desired detection probability with a significantly lower number of mobile nodes especially when the detection requirements are highly stringent. Similarly, when the target is mobile the coverage-based mobility model produces a consistently higher detection probability compared to other models under investigation.