We discuss the Monge problem for under-determined systems of ordinary differential equations with an arbitrary degree of freedom and give a sufficient condition, in terms of truncated multi-flag systems, for the Monge property to hold. This condition extends in a natural way the Cartan criterion valid for systems with one degree of freedom.

Let $d$ be a square-free positive integer and $l(d)$ be the period length of the simple continued fraction expansion of $\omega_d$, where $\omega_d $ is integral basis of $\mathbb{Z}[\sqrt{d}]$. Let $\varepsilon_d = (t_d+u_d\sqrt{d})/2$ $(>1)$ be the fundamental unit of the real quadratic field $\mathbb{Q}(\sqrt{d})$. In this paper new lower bounds for $\varepsilon_d$, $t_d$, and $u_d$ are described in terms of $l(d)$. The lower bounds of $\varepsilon_d$ are sharper than the known bounds and those of $t_d$ and $u_d$ have been yet unknown. In order to show the strength of the method of the proof, some interesting examples of $d$ are given for which $\varepsilon_d$ and Yokoi's $d$-invariants are determined explicitly in relation to continued fractions of the form $[a_0, \overline{1, \dots ,1, a_{l(d)}}\mkern1.5mu]$.

Wyner and Ziv (1989) studied the asymptotic properties of recurrence times of stationary ergodic processes, and applied the results to obtain optimal data compression schemes in information transmission. Since then many data compression algorithms based upon string matching of a sequence from an information source with a database have been proposed and studied. In this paper we consider Gaussian stationary processes representing an information source and a database, and study problems of string matching with distortion. We prove theorems concerning the asymptotic behavior of the probability of string matching with distortion and the waiting time for the string matching.

Consider $n$-th order linear differential equations with meromorphic periodic coefficients of the form $w^{(n)}+R_{n-1}(e^{z})w^{(n-1)}+\cdots+ R_{1}(e^{z})w'+R_{0}(e^{z})w = 0$, $n \ge 2$, where $R_{\nu}(t)$ $(0 \le \nu \le n-1)$ are rational functions of $t$. Under certain assumptions, we prove oscillation theorems concerning meromorphic solutions, which contain necessary conditions for the existence of a meromorphic solution with finite exponent of convergence of the zero-sequence. We also discuss meromorphic or entire solutions whose zero-sequences have an infinite exponent of convergence, and give a new zero-density estimate for such solutions.

We show that a continuous map of a metric space is pointwise recurrent and has the pseudo orbit tracing property if and only if the map is uniformly conjugate to an adding-machine-like map restricted to some invariant subset.

A birth and death process in a time-dependent random environment is introduced. We will discuss the recurrence and transience properties for the process.

We prove the invariance of plurigenera under smooth projective deformations in full generality.

We study the moduli space of stable sheaves on a reducible projective scheme by use of a suitable stratification of the moduli space. Each stratum is the moduli space of "triples"',which is the main object investigated in this paper. As an application, we can see that the relative moduli space of rank two stable sheaves on quadric surfaces gives a nontrivial example of the relative moduli space which is not flat over the base space.

We characterize numbers having purely periodic $\beta$-expansions where $\beta$ is a Pisot number satisfying a certain irreducible polynomial. The main tool of the proof is to construct a natural extension on a $d$-dimensional domain with a fractal boundary.