In this paper we introduce a notion of $q$-subharmonicity for non-smooth functions and then using $q$-subharmonic exhaustion function, define a $q$-pseudoconvexity which is applicable to the domain with non-smooth boundary. Among others, we generalize the Donnelly-Fefferamn type theorem on $q$-pseudoconvex domains and as an application of this theorem, we give approximation theorem for $\overline{\partial}$-closed forms.

We study the subalgebra of fixed points of a root graded Lie algebra under a certain class of finite order automorphisms. As the centerless core of extended affine Lie algebras or equivalently irreducible centerless Lie tori are examples of root graded Lie algebras, our work is an extension of some recent result about the subalgebra of fixed points of a Lie torus under a certain finite order automorphism.

We prove that the fundamental quandle of the trefoil knot is isomorphic to the projective primitive subquandle of transvections of the symplectic space $\mathbb{Z} \oplus \mathbb{Z}$. The last quandle can be identified with the Dehn quandle of the torus and the cord quandle on a 2-sphere with four punctures. We also show that the fundamental quandle of the long trefoil knot is isomorphic to the cord quandle on a 2-sphere with a hole and three punctures.

We consider the linear integro-differential operator $L$ defined by \begin{equation*} Lu(x) =\int_{\mathbb{R}^{n}}(u(x+y)-u(x) -\mathbbm{1}_{[1,2]}(\alpha)\mathbbm{1}_{\{|y|\leq 2\}}(y)y \cdot \nabla u(x))k(x,y)\, dy. \end{equation*} Here the kernel $k(x,y)$ behaves like $|y|^{-n-\alpha}$, $\alpha \in (0,2)$, for small $y$ and is Hölder-continuous in the first variable, precise definitions are given below. We study the unique solvability of the Cauchy problem corresponding to $L$. As an application we obtain well-posedness of the martingale problem for $L$. Our strategy follows the classical path of Stroock-Varadhan. The assumptions allow for cases that have not been dealt with so far.

Let $S$ be a Todorov surface, i.e., a minimal smooth surface of general type with $q=0$ and $p_{g}=1$ having an involution $i$ such that $S/i$ is birational to a $K3$ surface and such that the bicanonical map of $S$ is composed with $i$. The main result of this paper is that, if $P$ is the minimal smooth model of $S/i$, then $P$ is the minimal desingularization of a double cover of $\mathbb{P}^{2}$ ramified over two cubics. Furthermore it is also shown that, given a Todorov surface $S$, it is possible to construct Todorov surfaces $S_{j}$ with $K^{2}=1,\ldots,K_{S}^{2}-1$ and such that $P$ is also the smooth minimal model of $S_{j}/i_{j}$, where $i_{j}$ is the involution of $S_{j}$. Some examples are also given, namely an example different from the examples presented by Todorov in [9].

A new proof of the homogeneity of isoparametric hypersurfaces with six simple principal curvatures [4] is given in a method applicable to the multiplicity two case.

Let $\mathcal{T}(N)$ be the subgroup of the mapping class group of a nonorientable surface $N$ (possibly with punctures and/or boundary components) generated by twists about two-sided circles. We obtain a simple generating set for $\mathcal{T}(N)$. As an application we compute the first homology group (abelianization) of $\mathcal{T}(N)$.

We investigate the Cauchy problem for second order hyperbolic equations of complete form, and we prove an extension of a classical result of Oleĭnik [10] concerning the well-posedness for equations in which are absent the terms with mixed time-space derivatives. Then, in space dimension $n=1$, we compare our results with those in [8] for equations with analytic coefficients, and those of [7] and [11] for homogeneous equations with coefficients depending only either on $t$ or on $x$. Moreover we exhibit, in space dimension $n\ge 2$, an equation of the form \begin{equation*} u_{tt} - \sum_{i,j=1}^{n} (a_{ij}(t,x)u_{x_{j}})_{x_{i}} = 0{,} \quad\text{with}\quad \sum a_{ij} \xi_{i}\xi_{j} \ge 0, \end{equation*} where the coefficients are analytic functions, for which the Cauchy problem is ill-posed. Finally, we present a sufficient condition for the well-posedness of $2 \times 2$ systems.

In this paper, we investigate the behavior of the Miyazawa polynomial of periodic virtual links. As applications, we give some criteria to detect possible periods of a given oriented virtual link.

Let $\tilde{S}$ be a Riemann surface of type $(p,n)$ with $3p-3+n>0$. Let $F$ be a pseudo-Anosov map of $\tilde{S}$ defined by two filling simple closed geodesics on $\tilde{S}$. Let $a\in \tilde{S}$, and $S=\tilde{S} - \{a\}$. For any map $f\colon S\to S$ that is generated by two simple closed geodesics and is isotopic to $F$ on $\tilde{S}$, there corresponds to a configuration $\tau$ of invariant half planes in the universal covering space of $\tilde{S}$. We give a necessary and sufficient condition (with respect to the configuration) for those $f$ to be pseudo-Anosov maps. As a consequence, we obtain infinitely many pseudo-Anosov maps $f$ on $S$ that are isotopic to $F$ on $\tilde{S}$ as $a$ is filled in.

In this paper we classify completely all regular minimal surfaces with $K^{2}=8$, $p_{g}=4$ whose canonical map is composed with an involution. We obtain six unirational families. The last two are irreducible components of the moduli space of minimal surfaces of general type with $K^{2}=8$, $p_{g}=4$. These families hit three different topological types.

First, we generalize the big Picard theorem for pseudoholomorphic maps. Next, we prove Noguchi-type extension-convergence theorems for pseudoholomorphic maps. Finally, we show that the $J$-automorphism group of an almost complex submanifold hyperbolically embedded in a compact, almost complex manifold of real dimension 4 is finite when it is the complement of an immersed pseudoholomorphic curve.

We study a random measure which describes distribution of eigenvalues and corresponding eigenfunctions of random Schrödinger operators on $L^{2}(\mathbf{R}^{d})$. We show that in the natural scaling every limiting point is infinitely divisible.

We study the isentropic compressible Navier-Stokes equations with radially symmetric data and non-negative initial density in an annular domain. We prove the global existence of strong solutions for any $\gamma\geq 1$. Moreover, we obtain the uniform in time $L^{\infty}$-boundedness of the density and $H^{1}$-boundedness of the velocity, improving therefore the corresponding result in [2], where the condition $\gamma\geq 2$ is required to guarantee the existence.

We have two constructions of the level-$(0,1)$ irreducible representation of the quantum toroidal algebra of type $A$. One is due to Nakajima and Varagnolo-Vasserot. They constructed the representation on the direct sum of the equivariant K-groups of the quiver varieties of type $\hat{A}$. The other is due to Saito-Takemura-Uglov and Varagnolo-Vasserot. They constructed the representation on the q-deformed Fock space introduced by Kashiwara-Miwa-Stern. In this paper we give an explicit isomorphism between these two constructions. For this purpose we construct simultaneous eigenvectors on the q-Fock space using the nonsymmetric Macdonald polynomials. Then the isomorphism is given by corresponding these vectors to the torus fixed points on the quiver varieties.