Fast and Accurate Computation of Orbital Collision Probability for Short-Term Encounters

International audienceThis article provides a new method for computing the probability of collision between two spherical space objects involved in a short-term encounter under Gaussian-distributed uncertainty. In this model of conjunction, classical assumptions reduce the probability of collision to the integral of a two-dimensional Gaussian probability density function over a disk. The computational method presented here is based on an analytic expression for the integral, derived by use of Laplace transform and D-finite functions properties. The formula has the form of a product between an exponential term and a convergent power series with positive coefficients. Analytic bounds on the truncation error are also derived and are used to obtain a very accurate algorithm. Another contribution is the derivation of analytic bounds on the probability of collision itself, allowing for a very fast and - in most cases - very precise evaluation of the risk. The only other analytical method of the literature - based on an approximation - is shown to be a special case of the new formula. A numerical study illustrates the efficiency of the proposed algorithms on a broad variety of examples and favorably compares the approach to the other methods of the literature

The Tychonoff uniqueness theorem for the G-heat equation

In this paper, we obtain the Tychonoff uniqueness theorem for the G-heat equation

Uniqueness of nontrivially complete monotonicity for a class of functions involving polygamma functions

For $m,n\in\mathbb{N}$, let $f_{m,n}(x)=\bigr[\psi^{(m)}(x)\bigl]^2+\psi^{(n)}(x)$ on $(0,\infty)$. In the present paper, we prove using two methods that, among all $f_{m,n}(x)$ for $m,n\in\mathbb{N}$, only $f_{1,2}(x)$ is nontrivially completely monotonic on $(0,\infty)$. Accurately, the functions $f_{1,2}(x)$ and $f_{m,2n-1}(x)$ are completely monotonic on $(0,\infty)$, but the functions $f_{m,2n}(x)$ for $(m,n)\ne(1,1)$ are not monotonic and does not keep the same sign on $(0,\infty)$.Comment: 9 page

Full counting statistics of energy fluctuations in a driven quantum resonator

We consider the statistics of time-integrated energy fluctuations of a driven bosonic resonator (as measured by a QND detector), using the standard Keldysh prescription to define higher moments. We find that due to an effective cascading of fluctuations, these statistics are surprisingly non-classical: the low-temperature, quantum probability distribution is not equivalent to the high-temperature classical distribution evaluated at some effective temperature. Moreover, for a sufficiently large drive detuning and low temperatures, the Keldysh-ordered quasi-probability distribution characterizing these fluctuations fails to be positive-definite; this is similar to the full counting statistics of charge in superconducting systems. We argue that this indicates a kind of non-classical behaviour akin to that tested by Leggett-Garg inequalities.Comment: 10 pages, 2 figure

Operator solutions for fractional Fokker-Planck equations

We obtain exact results for fractional equations of Fokker-Planck type using evolution operator method. We employ exact forms of one-sided Levy stable distributions to generate a set of self-reproducing solutions. Explicit cases are reported and studied for various fractional order of derivatives, different initial conditions, and for different versions of Fokker-Planck operators.Comment: 4 pages, 3 figure

An exact analytical solution for generalized growth models driven by a Markovian dichotomic noise

Logistic growth models are recurrent in biology, epidemiology, market models, and neural and social networks. They find important applications in many other fields including laser modelling. In numerous realistic cases the growth rate undergoes stochastic fluctuations and we consider a growth model with a stochastic growth rate modelled via an asymmetric Markovian dichotomic noise. We find an exact analytical solution for the probability distribution providing a powerful tool with applications ranging from biology to astrophysics and laser physics

The Measure of the Orthogonal Polynomials Related to Fibonacci Chains: The Periodic Case

The spectral measure for the two families of orthogonal polynomial systems related to periodic chains with N-particle elementary unit and nearest neighbour harmonic interaction is computed using two different methods. The interest is in the orthogonal polynomials related to Fibonacci chains in the periodic approximation. The relation of the measure to appropriately defined Green's functions is established.Comment: 19 pages, TeX, 3 scanned figures, uuencoded file, original figures on request, some misprints corrected, tbp: J. Phys.

Influence of oxygen ordering kinetics on Raman and optical response in YBa_2Cu_3O_{6.4}

Kinetics of the optical and Raman response in YBa_2Cu_3O_{6.4} were studied during room temperature annealing following heat treatment. The superconducting T_c, dc resistivity, and low-energy optical conductivity recover slowly, implying a long relaxation time for the carrier density. Short relaxation times are observed for the B_{1g} Raman scattering -- magnetic, continuum, and phonon -- and the charge transfer band. Monte Carlo simulations suggest that these two relaxation rates are related to two length scales corresponding to local oxygen ordering (fast) and long chain and twin formation (slow).Comment: REVTeX, 3 pages + 4 PostScript (compressed) figure

A taste of the deep-sea: The roles of gustatory and tactile searching behaviour in the grenadier fish <i>Coryphaenoides armatus</i>

The deep-sea grenadier fishes (Coryphaenoides spp.) are among the dominant predators and scavengers in the ocean basins that cover much of Earth's surface. Baited camera experiments were used to study the behaviour of these fishes. Despite the apparent advantages of rapidly consuming food, grenadiers attracted to bait spend a large proportion of their time in prolonged periods of non-feeding activity. Video analysis revealed that fish often adopted a head-down swimming attitude (mean of 21.3 degrees between the fish and seafloor), with swimming velocity negatively related to attitude. The fish also swam around and along vertical and horizontal structures of the lander with their head immediately adjacent to the structure. We initially hypothesised that this behaviour was associated with the use of the short chin barbel in foraging. Barbel histology showed numerous taste buds in the skin, and a barbel nerve with about 20,000 axons in adult fish. A tracing experiment in one undamaged animal revealed the termination fields of the barbel neurons in the trigeminal and rhombencephalic regions, indicating both a mechanoreceptory and a gustatory role for the barbel. Our conclusion was that olfactory foraging becomes ineffective at close ranges and is followed by a search phase using tactile and gustatory sensing by the barbel. The development of this sensory method probably co-evolved alongside behavioural changes in swimming mechanics to allow postural stability at low swimming speeds