In this paper, the comparison theorem of stochastic differential delay equations with Markovian switching has been investigated. In order to develop our theory, a new proof of existence and uniqueness of stochastic differential equations with Markovian switching is given.

One of the earliest and best-known application of the probabilistic method is the proof of existence of a 2 logn-Ramsey graph, i.e., a graph with n nodes that contains no clique or independent set of size 2 logn. The explicit construction of such a graph is a major open problem. We show that a reasonable hardness assumption implies that in polynomial time one can construct a list containing pol...

Using a weighted Hs-contraction mapping argument based on the macro-micro decomposition of Liu and Yu, we give an elementary proof of existence, with sharp rates of decay and distance from the Chapman–Enskog approximation, of small-amplitude shock profiles of the Boltzmann equation with hard-sphere potential, recovering and slightly sharpening results obtained by Caflisch and Nicolaenko using d...

Using language of dg PROPs we give a new short proof of existence of star products on (formal) germs of Poisson manifolds.

We study a novel two–sided Stefan problem – motivated by the study of certain 2D interfaces – in which boundaries at both sides of the sample encroach into the bulk with rate equal to the boundary value of the gradient. Here the density is in [0, 1] and takes the two extreme values at the two free boundaries. It is noted that the problem is borderline ill–posed: densities in excess of unity lia...

We investigate the band-gap structure of the spectrum of second-order partial differential operators associated with the propagation of waves in a periodic two-component medium. The medium is characterized by a real-valued position-dependent periodic function "(x) that is the dielectric constant for electromagnetic waves and mass density for acoustic waves. The imbedded component consists of a ...

In this paper the notion of uniier is extended to the innnite set case. The proof of existence of the most general uniier of any innnite, uniiable set of types (terms) is presented. Learning procedure, based on innnite set uniication, is described.