Let $X$ be a Feller process that has strong Feller property. In this paper, we investigate the Feller as well as strong Feller properties of the semigroups generated by multiplicative functionals of $X$ in open sets. Special attention is given to the Feynman-Kac and Girsanov transforms of $X$. Three examples of local Kato class measure that are not of Kato class are given in the last section so that Feller and strong Feller properties hold for corresponding Feynman-Kac semigroup of $X$ in open sets.

We study a one-dimensional fluid dynamical model of semiconductors. Our goal in this paper is to prove the uniqueness of stationary solutions.

We study Fox 5-colorings for diagrams of 1- and 2-dimensional knots. We prove that any 5-colorable 1-knot has a non-trivially 5-colored diagram such that exactly four colors of five are assigned to the arcs of the diagram. Moreover, we prove that there is a 5-colorable 2-knot such that, for any non-trivially 5-colored diagram, all five colors are assigned to the sheets of the diagram.

In a previous paper [4], we introduced the notion of real quadratic fields with period $l$ of minimal type in terms of continued fractions. As a consequence, we have to examine a construction of real quadratic fields with period $\ge 5$ of minimal type in order to find many real quadratic fields of class number 1. When $l \ge 4$, it appears that there exist infinitely many real quadratic fields with period $l$ of minimal type. Indeed, we provided an infinitude of real quadratic fields with period 4 of minimal type in [4]. In this paper, we construct an infinite family of real quadratic fields with large even period of minimal type whose class number is greater than any given positive integer, and whose Yokoi invariant is greater than any given positive integer.

The Chebyshev map is a typical chaotic map. We consider generalized Chebyshev maps $T_{k}$ of $\mathbf{C}^{2}$. The support of the maximal entropy measure of $T_{k}$ is connected. We perturb $T_{k}$ in a certain direction. Then we can show the support of the maximal entropy measure of this map is a Cantor set.

Following a suggestion made by J.-P. Demailly, for each $k\ge 1$, we endow, by an induction process, the $k$-th (anti)tautological line bundle $\mathcal{O}_{X_{k}}(1)$ of an arbitrary complex directed manifold $(X,V)$ with a natural smooth Hermitian metric. Then, we compute recursively the Chern curvature form for this metric, and we show that it depends (asymptotically---in a sense to be specified later) only on the curvature of $V$ and on the structure of the fibration $X_{k}\to X$. When $X$ is a surface and $V=T_{X}$, we give explicit formulae to write down the above curvature as a product of matrices. As an application, we obtain a new proof of the existence of global invariant jet differentials vanishing on an ample divisor, for $X$ a minimal surface of general type whose Chern classes satisfy certain inequalities, without using a strong vanishing theorem [1] of Bogomolov.

In this paper we give a generalization of a result of Herzog, Hibi, and Zheng providing an upper bound for regularity of powers of an ideal. As the main result of the paper, we give a simple criterion in terms of Rees algebra of a given ideal to show that high enough powers of this ideal have linear resolution. We apply the criterion to two important ideals $J,J_{1}$ for which we show that $J^{k}$, and $J_{1}^{k}$ have linear resolution if and only if $k\neq 2$. The procedures we include in this work is encoded in computer algebra package CoCoA [3].

We give the best possible upper bound on the number of exceptional values and the totally ramified value number of the hyperbolic Gauss map for pseudo-algebraic constant mean curvature one surfaces in the hyperbolic three-space and some partial results on the Osserman problem for algebraic case. Moreover, we study the value distribution of the hyperbolic Gauss map for complete constant mean curvature one faces in de Sitter three-space.

We consider closed orientable 3-dimensional hyperbolic manifolds which are cyclic branched coverings of the 3-sphere, with branching set being a two-bridge knot (or link). We establish two-sided linear bounds depending on the order of the covering for the Matveev complexity of the covering manifold. The lower estimate uses the hyperbolic volume and results of Cao-Meyerhoff, Guéritaud-Futer (who recently improved previous work of Lackenby), and Futer-Kalfagianni-Purcell, and it comes in two versions: a weaker general form and a shaper form. The upper estimate is based on an explicit triangulation, which also allows us to give a bound on the Delzant T-invariant of the fundamental group of the manifold.

In this paper, we generalise the first Klein-Maskit combination theorem to discrete groups of Möbius transformations in higher dimensions. The application of the main theorem is discussed in the last section.

Let $f\colon \mathbb{G}_{n,k}\to \mathbb{G}_{m,l}$ be any continuous map between two distinct complex (resp. quaternionic) Grassmann manifolds of the same dimension. We show that the degree of $f$ is zero provided $n,m$ are sufficiently large and $l\geq 2$. If the degree of $f$ is $\pm 1$, we show that $(m,l)=(n,k)$ and $f$ is a homotopy equivalence. Also, we prove that the image under $f^{*}$ of every element of a set of algebra generators of $H^{*}(\mathbb{G}_{m,l};\mathbb{Q})$ is determined up to a sign, $\pm$, by the degree of $f$, provided this degree is non-zero.

We systematically study quasi-sections of morphisms of log schemes. As applica-tions, we prove exact log flat descents. Our main tool is the structure theorem of $\mathbb{Q}$-integral homomorphisms of monoids.