An interplay between attraction and repulsion in infinite populations

We propose and study a model describing an infinite population of point entities arriving in and departing from $$X=\mathbb {R}^d$$ X = R d , $$d\ge 1$$ d ≥ 1 . The already existing entities force each other to leave the population (repulsion) and attract the newcomers. The evolution of the population states is obtained by solving the corresponding Fokker-Planck equation. Without interactions, the evolution preserves states in which the probability $$p(n,\Lambda )$$ p ( n , Λ ) of finding n points in a compact vessel $$\Lambda \subset X$$ Λ ⊂ X obeys the Poisson law. As we show, for pure attraction the decay of $$p(n,\Lambda )$$ p ( n , Λ ) with $$n\rightarrow +\infty $$ n → + ∞ may be essentially slower. The main result is the statement that in the presence of repulsion—even of an arbitrary short range—the evolution preserves states in which the decay of $$p(n,\Lambda )$$ p ( n , Λ ) is at most Poissonian. We also derive the corresponding kinetic equation, the numerical solutions of which can provide more detailed information on the interplay between attraction and repulsion. Further possibilities in studying the proposed model are also discussed.

Extreme Value Theory for Hurwitz Complex Continued Fractions

The Hurwitz complex continued fraction is a generalization of the nearest integer continued fraction. In this paper, we prove various results concerning extremes of the modulus of Hurwitz complex continued fraction digits. This includes a Poisson law and an extreme value law. The results are based on cusp estimates of the invariant measure about which information is still limited. In the process, we obtained several results concerning the extremes of nearest integer continued fractions as well.

On Approximation of the Tails of the Binomial Distribution with These of the Poisson Law

A subject of this study is the behavior of the tail of the binomial distribution in the case of the Poisson approximation. The deviation from unit of the ratio of the tail of the binomial distribution and that of the Poisson distribution, multiplied by the correction factor, is estimated. A new type of approximation is introduced when the parameter of the approximating Poisson law depends on the point at which the approximation is performed. Then the transition to the approximation by the Poisson law with the parameter equal to the mathematical expectation of the approximated binomial law is carried out. In both cases error estimates are obtained. A number of conjectures are made about the refinement of the known estimates for the Kolmogorov distance between binomial and Poisson distributions.

On a generalization of the Rényi–Srivastava characterization of the Poisson law

We give a new method of proof for a result of D. Pierre-Loti-Viaud and P. Boulongne which can be seen as a generalization of a characterization of Poisson law due to Rényi and Srivastava. We also provide explicit formulas, in terms of Bell polynomials, for the moments of the compound distributions occurring in the extended collective model in non-life insurance.

FORMULATION OF THE PROBLEM OF PLANNING AIRCRAFT ROUTES WHEN SERVICING A RANDOM FLOW OF APPLICATIONS ARRIVING IN FLIGHT

The work is about process of ground facilities servicing by aircrafts group when receiving requests during the flight. In case of manned aircraft, servicing of one request refers to the flight from the point to another one. As for unmanned aircrafts, it is necessary to observe reached destination point at the specified speed and flight altitude. It is assumed that the distance between the points and the moments of requests appearing ranked by the Poisson law. Necessary condition of successful task solving is that the average service speed of the requests flow exceeds the average speed of their receipt. It was found that taking into account the randomness of requests, three flight situations are possible – the “idle” mode in service when the requests number is less than the number of unoccupied aircraft, the regular mode when the balance of supply and demand and the “peak” mode when the requests number is increased and there is a queue in service.

The shape of the photon transfer curve of CCD sensors

The photon transfer curve (PTC) of a CCD depicts the variance of uniform images as a function of their average. It is now well established that the variance is not proportional to the average, as Poisson statistics would indicate, but rather flattens out at high flux. This “variance deficit”, related to the brighter-fatter effect, feeds correlations between nearby pixels that increase with flux, and decay with distance. We propose an analytical expression for the PTC shape, and for the dependence of correlations with intensity, and relate both to some more basic quantities related to the electrostatics of the sensor, which are commonly used to correct science images for the brighter-fatter effect. We derive electrostatic constraints from a large set of flat field images acquired with a CCD e2v 250, and eventually question the generally-admitted assumption that boundaries of CCD pixels shift by amounts proportional to the source charges. Our results show that the departure of flat field statistics from the Poisson law is entirely compatible with charge redistribution during the drift in the sensor.

Limit Poisson law for the distribution of the number of components in generalized allocation scheme

We consider problems on the convergence of distributions of the total number of components and numbers of components with given volume to the Poisson law. Sufficient conditions of such convergence are given. Our results generalize known statemets on the limit Poisson laws of the number of components (cycles, unrooted and rooted trees, blocks and other structures) in the corresponding generalized of allocation schemes.