We develop the theory of cohomology groups for recurrence relations, based upon the asymptotic analysis of finite difference equations carried out in a previous paper. We apply it to compute the Gevrey extension groups of the $\mathcal{D}$-modules associated to some confluent hypergeometric systems. In those applications, recurrence relations appear as contiguity relations of hypergeometric systems.

We denote by $HP\left(R\right)$ and ($HB\left(R\right)$, resp.) the class of positive (bounded, resp.) harmonic functions on a Riemann surface $R$. Consider an open Riemann surface $W$ possessing a Green's function and a $p$-sheeted unlimited covering surface $\tilde{W}$ of $W$ with projection map $\varphi$. We give a necessary and sufficient condition, in terms of Martin boundary, for $HX\left(W\right)\circ$$\varphi$$=HX\left(\tilde{W}\right)(X=P,B)$. We also give some examples illustrating the above result when $W$ is the unit disc.

We investigate surfaces of constant mean curvature with a nontrivial fundamental group in a Weierstraß type representation. As an application we construct a family of complete CMC immersions which are conformally cylinders and have umbilics.

We use cohomological group theory and properties of $L^2$-Betti numbers to determine the solvable groups with presentations of deficiency 1, to give a new proof of the "Tits alternative" for subgroups of Haken 3-manifold groups, and to study the fundamental groups of closed 4-manifolds with Euler characteristic 0 and, in particular, 2-knot groups.

Given a quasi-subaditive operator $T:L_0\left(\mu \right)$$\to L_0\left(v\right)$, we want to study mapping properties of interpolation type for which the following modular inequality holds

${\int }_{N}P\left(\right|Tf\left(x\right)\left|\right)dv\left(x\right)\le {\int }_{M}Q\left(\right|f\left(x\right)\left|\right)d\mu \left(x\right)$

where $P$ and $Q$ are modular functions. These results generalize the Marcinkiewicz interpolation theorem.

We examine plane waves of the elastic reduced wave equation in the half-space, and by linear combinations of them we construct the solution coinciding with any plane wave on the boundary. In the (dual) variable in the normal direction to the boundary, we apply the complex analysis to the inverse matrix of the elastic symbol.

It is proved that pointwise commuting formally normal operators which are dominated by a single essentially normal operator are essentially normal and essentially spectrally commuting. The question when essential normality of a polynomial in an operator implies essential normality of that operator is solved in this way. Furthermore, domination by essentially normal powers of formally normal operators are studied and, as a consequence, extended versions of Nelson's criterion for essential spectral commutativity are proposed. Subsequent domination results ensuring joint subnormality of systems of operators are proved. Several applications to multidimensional moment problems are found.

We define Julia sets for (topological) expanding postcritically finite branched coverings on $S^2$, and show the existence and the uniqueness of Julia sets. Our main aim is the investigation of codings of Julia sets (i.e. semiconjugacies between symbolic dynamics and Julia sets). In particular, it is proved that if two expanding branched coverings are combinatorially equivalent, then their Julia sets are topologically conjugate.

We prove that for an expanding postcritically finite branched covering $f$, the Julia set is orientedly $S^1$-parametrizable if and only if $f^n$ is combinatorially equivalent to the degenerate mating of two polynomials for some $n>0$.

This paper is concerned with the standard $L_p$ estimate of solutions to the resolvent problem for the Stokes operator on an infinite layer.

We introduce lower and upper semi-continuity of a map to the Banach space $c_0$($\lambda$) for an infinite cardinal $\lambda$. We prove that the following conditions (i), (ii) and (iii) on a $T_1$-space $X$ are equivalent: (i) For every two maps $g,$$h$ : $X\to c_0$($\lambda$) such that $g$ is upper semi-continuous, $h$ is lower semi-continuous and $g\le h$, there exists a continuous map $f$ : $X\to c_0\left(\lambda \right)$, with $g\le f\le h$. (ii) For every Banach space $Y$, with $w\left(Y\right)\le \lambda ,$ every lower semi-continuous set-valued mapping $\varnothing$ : $X\to {\mathcal{C}}_c\left(Y\right)$ admits a continuous selection, where ${\mathcal{C}}_c\left(Y\right)$ is the set of all non-empty compact convex sets in Y. (iii)$X$ is normal and every locally finite family $F$ of subsets of $X$, with $\left|F\right|\le \lambda$, has a locally finite open expansion provided it has a point-finite open expansion. We also characterize several paracompact-like properties by inserting continuous maps between semi-continuous Banach-valued functions.

We have developed the theory of $K{M}_{2}O$-Langevin equations for stationary and non-degenerate flow in an inner product space. As its generalization and refinement of the results in [14], [15], [16], we shall treat in this paper a general flow in an inner product space without both the stationarity property and the non-degeneracy property. At first, we shall derive the $K{M}_{2}O$-Langevin equation describing the time evolution of the flow and prove the fluctuation-dissipation theorem which states that there exists a relation between the fluctuation part and the dissipation part of the above $K{M}_{2}O\text{-}$ Langevin equation. Next we shall prove the characterization theorem of stationarity property, the construction theorem of a flow with any given nonnegative definite matrix function as its two-point covariance matrix function and the extension theorem of a stationary flow without losing stationarity property.