Let $R_{n} (x)$ be the first return time of the initial sequence $x_{1} \cdots x_{n}$ of $x = x_{1} x_{2} \cdots$. For mixing processes, sharp bounds for the convergence of $R_{n} (x) P_{n}(x)$ to exponential distribution are presented, where $P_{n} (x)$ is the probability of $x_{1} \cdots x_{n}$. As a corollary, the limit of the mean of $\log(R_{n}(x) P_{n}(x))$ is obtained. For exponentially $\phi$-mixing processes, $-E[\log(R_{n} P_{n})]$ converges exponentially to the Euler's constant. A similar result is observed for the hitting time.

Let $R$ be a left and right Noetherian ring. We introduce the notion of the torsionfree dimension of finitely generated $R$-modules. For any $n \geq 0$, we prove that $R$ is a Gorenstein ring with self-injective dimension at most $n$ if and only if every finitely generated left $R$-module and every finitely generated right $R$-module have torsionfree dimension at most $n$, if and only if every finitely generated left (or right) $R$-module has Gorenstein dimension at most $n$. For any $n \geq 1$, we study the properties of the finitely generated $R$-modules $M$ with $\Ext_{R}^{i}(M, R)=0$ for any $1 \leq i \leq n$. Then we investigate the relation between these properties and the self-injective dimension of $R$.

We study the asymptotic stability of nonlinear waves for damped wave equations with a convection term on the half line. In the case where the convection term satisfies the convex and sub-characteristic conditions, it is known by the work of Ueda [7] and Ueda--Nakamura--Kawashima [10] that the solution tends toward a stationary solution. In this paper, we prove that even for a quite wide class of the convection term, such a linear superposition of the stationary solution and the rarefaction wave is asymptotically stable. Moreover, in the case where the solution tends to the non-degenerate stationary wave, we derive that the time convergence rate is polynomially (resp. exponentially) fast if the initial perturbation decays polynomially (resp. exponentially) as $x \to \infty$. Our proofs are based on a technical $L^{2}$ weighted energy method.

We give examples of degree functions $\deg\colon R \to M \cup \{-\infty\}$, where $R$ is $\mathbb{C}[X,Y]$ or $\mathbb{C}[X,Y,Z]$ and $M$ is $\mathbb{Z}$ or $\mathbb{N}$, whose behaviour with respect to $\mathbb{C}$-derivations $D\colon R \to R$ is pathological in the sense that $\{\deg(Dx) - \deg(x) \mid x \in R\setminus \{0\}\}$ is not bounded above. We also give several general results stating that such pathologies do not occur when the degree functions satisfy certain hypotheses.

We study the structure of Stanley--Reisner rings associated to cyclic polytopes, using ideas from unprojection theory. Consider the boundary simplicial complex $\Delta(d,m)$ of the $d$-dimensional cyclic polytope with $m$ vertices. We show how to express the Stanley--Reisner ring of $\Delta(d,m+1)$ in terms of the Stanley--Reisner rings of $\Delta(d,m)$ and $\Delta(d-2,m-1)$. As an application, we use the Kustin--Miller complex construction to identify the minimal graded free resolutions of these rings. In particular, we recover results of Schenzel, Terai and Hibi about their graded Betti numbers.

In this paper, we consider type-changing equations for one unknown function of two variables by using the theory of differential systems. We give fundamental properties and provide a notion of geometric solutions from a viewpoint of contact geometry of second order. Moreover, we study the structure of associated overdetermined systems and obtain an existence condition of solutions of a special class which are called parabolic solutions of type-changing equations.

Let $Q^{m+1}$ denote the family of regular Sasakian manifolds whose base manifold $M^{2m}$ is a compact symmetric space. We provide a classification of the totally geodesic submanifolds of $Q^{m+1}$ which are invariant, anti-invariant of maximal dimension or contact CR with respect to the Sasakian structure. Such submanifolds are closely related to complex and totally real totally geodesic submanifolds of the Hermitian symmetric space $M^{2m}$.

Ozeki and Takeuchi [14] introduced the notion of Condition A and Condition B to construct two classes of inhomogeneous isoparametric hypersurfaces with four principal curvatures in spheres, which were later generalized by Ferus, Karcher and Münzner to many more examples via the Clifford representations; we will refer to these examples of Ozeki and Takeuchi and of Ferus, Karcher and Münzner collectively as OT-FKM type throughout the paper. Dorfmeister and Neher [5] then employed isoparametric triple systems [3, 4], which are algebraic in nature, to prove that Condition A alone implies the isoparametric hypersurface is of OT-FKM type. Their proof for the case of multiplicity pairs $\{3, 4\}$ and $\{7, 8\}$ rests on a fairly involved algebraic classification result [9] about composition triples. In light of the classification [2] that leaves only the four exceptional multiplicity pairs $\{4, 5\}, \{3, 4\}, \{7, 8\}$ and $\{6, 9\}$ unsettled, it appears that Condition A may hold the key to the classification when the multiplicity pairs are $\{3, 4\}$ and $\{7, 8\}$. Thus Condition A deserves to be scrutinized and understood more thoroughly from different angles. In this paper, we give a fairly short and rather straightforward proof of the result of Dorfmeister and Neher, with emphasis on the multiplicity pairs $\{3, 4\}$ and $\{7, 8\}$, based on more geometric considerations. We make it explicit and apparent that the octonion algebra governs the underlying isoparametric structure.

We introduce an elementary way of constructing principal $(\mathbb{Z}_{2})^{m}$-bundles over compact smooth manifolds. In addition, we will define a general notion of locally standard $(\mathbb{Z}_{2})^{m}$-actions on closed manifolds for all $m \geq 1$, and then give a general way to construct all such $(\mathbb{Z}_{2})^{m}$-actions from the orbit space. Some related topology problems are also studied.

We introduce and study knotoids. Knotoids are represented by diagrams in a surface which differ from the usual knot diagrams in that the underlying curve is a segment rather than a circle. Knotoid diagrams are considered up to Reidemeister moves applied away from the endpoints of the underlying segment. We show that knotoids in $S^{2}$ generalize knots in $S^{3}$ and study the semigroup of knotoids. We also discuss applications to knots and invariants of knotoids.

For normal two dimensional hypersurface singularities of Brieskorn type, concrete descriptions are given to both the fundamental cycle and the maximal ideal cycle on a star-shaped good resolution space. It is determined when these two cycles coincide.

For each closed oriented $3$-manifold $M$ in Thurston's picture, the set of degrees of self-maps on $M$ is given.