Diffusion with interactions and collisions between coloured particles and the propagation of chaos

Summary One considers a system ofN particles on the real line which are of two different types (colours). Their dynamics is given by a stochastic differential equation with constant diffusion part; the drift felt by a particle of either type depends on the empirical measures of type 1 and 2 particles at every instant; further a reflection condition is imposed so that particles of different type are not allowed to cross each other. The article studies the Vlasov-McKean limit of the system asN: propagation of chaos and an evolution equation for the limiting empirical measures is established, from where in particular an equation for the separating front between the two types follos.To the Memory of the late Professor Gishiro Maruýama     

Critical decomposable Bellman-Harris processes with two types of particles

Translated from Matematicheskie Zametki, Vol. 43, No. 2, pp. 276–282, February, 1988.     

Uniqueness of diffusion generators for two types of particle systems with singular interactions

For two types of stochastic particle systems in we show non-explosion in finite time by proving that their respective generators are L¹(μ)-unique, where μ is their respective invariant (in these cases even symmetrizing) measure. We also prove the much harder L²(μ)-uniqueness in both models.     

A conditional functional limit theorem for decomposable branching processes with two types of particles

Consider a critical decomposable branching process with two types of particles in which particles of the first type give birth, at the end of their life, to descendants of the first type, as well as to descendants of the second type, while particles of the second type produce only descendants of the same type at the time of their death. We prove a functional limit theorem describing the distribution for the total number of particles of the second type appearing in the process in time Nt, 0 ≤ t < ∞, given that the number of particles of the first type appearing in the process during its evolution is N.     

Relations between two types of polarized partitions

It is shown that close links exist between two important types of polarized partition relations of higher dimensions. Project supported by a South-South Fellowship of the Third World Academy of Sciences and by the National Natural Science Foundation of China (Grant No. 19671061).     

On entire solutions of two types of systems of complex differential-difference equations

In this paper, we will mainly investigate entire solutions with finite order of two types of systems of differential-difference equations, and obtain some interesting results. It extends some results concerning complex differential (difference) equations to the systems of differential-difference equations.     

Behavioral relationships between two nonlinear integrodifferential equations

In this paper the stability behavior of the solutions of a nonlinear system of integrodifferential equations and the solutions of a perturbation of this system is compared. Integral inequalities is the main technique employed.     

Maximal branching processes with several types of particles

The paper introduces multitype maximal branching processes and their generalizations. Some examples of their explicit construction are given. The ergodic theorem is proved for the case of two types of particles.     

Exact solutions of the equations of two-phase dynamics. Collapse of gas and particles in space

Under consideration is the system of partial differential equations describing the dynamics of a two-phase medium. Exact partially invariant solutions of rank 1 and defect 1 of this system are obtained with respect to some four-dimensional subalgebras. The phenomenon of collapse (an instantaneous source) in a two-phase medium is described.     

Occupancy with two types of balls

The classical occupancy problem is extended to the case where two types of balls are thrown. In particular, the probability that no urn contains both types of balls is studied. This is a birthday problem in two groups of boys and girls to consider the coincidence of a boy's and a girl's birthday. Let N ₁ and N ₂ denote the numbers of balls of each type thrown one by one when the first collision between the two types occurs in one of m urns. Then N ₁ N ₂/m is asymptotically exponentially distributed as m tends to infinity.This problem is related to the security evaluation of authentication procedures in electronic message communication.     

Statistical mechanics of classical particles with logarithmic interactions

The inhomogeneous mean-field thermodynamic limit is constructed and evaluated for both the canonical thermodynamic functions and the states of systems of classical point particles with logarithmic interactions in two space dimensions. The results apply to various physical models of translation invariant plasmas, gravitating systems, as well as to planar fluid vortex motion. For attractive interactions a critical behavior occurs which can be classified as an extreme case of a second-order phase transition. To include in particular attractive interactions a new inequality for configurational integrals is derived from the arithmetic-geometric mean inequality. The method developed in this paper is easily seen to apply as well to systems with fairly general interactions in all space dimensions. In addition, it also provides us with a new proof of the Trudinger-Moser inequality known from differential geometry – in its sharp form.     

The relationships between some types of partial differential equations and ordinary differential equations as well as their applications

In this article, we establish some relationships between several types of partial differential equations and ordinary differential equations. One application of these relationships is that we can get the exact values of the blowup time and the blowup rate of the solution to a partial differential equation by solving an ordinary differential equation. Another application of these relationships is that we can give the estimates for the spatial integration (or mean value) of the solution to a partial differential equation. We also obtain the lower and upper bounds for the blowup time of the solution to a parabolic equation with weighted function and space‐time integral in the nonlinear term.