We introduce the pseudo-Einstein structure on real hypersurfaces in a K\"ahlerian manifold, namely, the Ricci curvature tensor for the generalized Tanaka-Webster connection (restricted) on the Levi subbundle $D$ is proportional to the Levi form. In particular, we give a classification of pseudo-Einstein Hopf-hypersurfaces in a non-flat complex space form.

In this paper, we present some class of three dimensional $C^{\infty}$ diffeomorphisms with nondegenerate one-sided homoclinic tangencies $q$ associated with hyperbolic fixed points $p$ each of which exhibits a horseshoe set. A key point in the proof is the existence of a transverse homoclinic point arbitrarily close to $q$. This result together with Birkhoff-Smale Theorem implies the existence of a horseshoe set arbitrarily close to $q$.

The concept of Chu correspondences between formal contexts is introduced. The construction of formal concepts induces a functor $Gal$ from the category of Chu correspondences to the category of sup-preserving maps between complete lattices. It turns out that the category of Chu correspondences has a $*$-autonomous category structure and the functor $Gal$ is shown to preserve the $*$-autonomous category structure.

This paper studies time regularity for the random walk governed by a probability measure $\mu$ on a locally compact, compactly generated group $G$. If $\mu$ is eventually coset aperiodic on $G$ and satisfies certain additional conditions, we establish that the associated Markov operator $T_{\mu}$ is analytic in $L^2(G)$, that is, one has an estimate $\|(I-T_{\mu}) T_{\mu}^n \| \leq cn^{-1}$, $n\in \mathbb{N}$, in $L^2$ operator norm. Alternatively, if $\mu$ is irreducible with period $d$ and satisfies certain conditions, we show that $T_{\mu}^d$ is analytic in $L^2(G)$. To obtain these results, we develop a number of interesting algebraic and spectral properties of coset aperiodic or irreducible measures on groups.

Let $S_t:M \to M$ be a semi-flow preserving a probability measure $\mu$ on a compact manifold $M$ and $g:M\to\mathbb{R}$ a real-valued Borel measurable function. We show a generalized local limit theorem for the stationary process $\{g\circ S_t;t\geq 0\}$ on the probability space $(M,\mu)$ under certain conditions.

Using the notion of weighted sharing of values we prove some uniqueness theorems for meromorphic functions which improve some existing results.

In the context of an infinite network $N$, the Dirichlet problem with respect to an arbitrary subset of vertices $N$ is solved.Using this solution, some of the important potential-theoretic concepts like Balayage, Domination principle, and Poisson kernel are investigated in $N$.

In this paper, we study the pro-$\Sigma$ fundamental groups of configuration, where $\Sigma$ is either the set of all prime numbers or a set consisting of a single prime number. In particular, we show, via two somewhat distinct approaches, that, in many cases, the "fiber subgroups" of such fundamental groups arising from the various natural projections of a configuration space to lower-dimensional configuration spaces may be characterized group-theoretically.