Up to a finite étale covering, we classify smooth projective $3$-folds $X$ with $\kappa(X) = -\infty$ admitting a nonisomorphic étale endomorphism in the case where there exists an FESP $Y_{\bullet}$ constructed from $X$ by a sequence of blowing-downs of an ESP and an extremal ray $R_{\bullet}$ of fiber type on $\operatorname{\overline{NE}}(Y_{\bullet})$ such that the pair $(Y_{\bullet}, R_{\bullet})$ is of type $(\text{C}_1)$ or $(\text{C}_0)$.

We analyse analytic properties of nonlocal transition semigroups associated with a class of stochastic differential equations (SDEs) in ${\mathbb{R}}^d$ driven by pure jump-type Lévy processes. First, we will show under which conditions the semigroup will be analytic on the Besov space $B_{p,q}^ m({\mathbb{R}}^d)$ with $1\le p, q\lt\infty$ and $m\in{\mathbb{R}}$. Secondly, we present some applications by proving the strong Feller property and give weak error estimates for approximating schemes of the SDEs over the Besov space $B_{\infty,\infty}^ m({\mathbb{R}}^d)$. The choice of Besov spaces is twofold. First, observe that Besov spaces can be defined via the Fourier transform and the partition of unity. Secondly, the space of continuous functions can be characterised by Besov spaces.

In this article we prove the following boundedness result: Fix a DCC set $I\subset [0, 1]$. Let $\mathfrak{D}$ be the set of all log pairs $(X, \Delta)$ satisfying the following properties: (i) $X$ is a projective surface defined over an algebraically closed field, (ii) $(X, \Delta)$ is log canonical and the coefficients of $\Delta$ are in $I$, and (iii) $K_X+\Delta$ is big. Then there is a positive integer $N=N(I)$ depending only on the set $I$ such that the linear system $|\lfloor m(K_X+\Delta)\rfloor|$ defines a birational map onto its image for all $m\geq N$ and $(X, \Delta)\in\mathfrak{D}$.

Let $(X^n, \check{X}^n)$ be a mirror pair of an $n$-dimensional complex torus $X^n$ and its mirror partner $\check{X}^n$. Then, a simple projectively flat bundle $E(L,\mathcal{L})\rightarrow X^n$ is constructed from each affine Lagrangian submanifold $L$ in $\check{X}^n$ with a unitary local system $\mathcal{L}\rightarrow L$. In this paper, we first interpret these simple projectively flat bundles $E(L,\mathcal{L})$ in the language of factors of automorphy. Furthermore, we give a geometric interpretation for exact triangles consisting of three simple projectively flat bundles $E(L,\mathcal{L})$ and their shifts by focusing on the dimension of intersections of the corresponding affine Lagrangian submanifolds $L$. Finally, as an application of this geometric interpretation, we discuss whether such an exact triangle on $X^n$ ($n\geq 2$) is obtained as the pullback of an exact triangle on $X^1$ by a suitable holomorphic projection $X^n\rightarrow X^1$.

Improving a theorem in [1] we observe that a maximal torus of an extrinsic symmetric space in a euclidean space $V$ is itself extrinsic symmetric in some affine subspace of $V$.

We introduce a concept of connectedness of antipodal sets of compact Riemannian symmetric spaces and construct a method to make a bigger antipodal set from a given antipodal set. Moreover, using the connectedness we give a sufficient condition that a given maximal antipodal set is homogeneous.

We classify all non-simple connected quandles of size $pq$ and $4p$ where $p,q$ are primes, as a special family of locally strictly simple quandles (i.e. quandles for which all proper subquandles are strictly simple). In particular, we classify all latin quandles of size $pq$ and $4p$ and we show that latin quandles of size $8p$ are affine.

The cohomological rigidity problem for toric manifolds asks whether toric manifolds are diffeomorphic (or homeomorphic) if their integral cohomology rings are isomorphic. Many affirmative partial solutions to the problem have been obtained and no counterexample is known. In this paper, we study the diffeomorphism classification of toric Fano $d$-folds with $d=3,4$ or with Picard number $\ge 2d-2$. In particular, we show that those manifolds except for two toric Fano $4$-folds are diffeomorphic if their integral cohomology rings are isomorphic. The exceptional two toric Fano $4$-folds (their ID numbers are 50 and 57 on a list of Øbro) have isomorphic cohomology rings and their total Pontryagin classes are preserved under an isomorphism between their cohomology rings, but we do not know whether they are diffeomorphic or homeomorphic.

Let $L$ be a positive-definite, even, integral lattice and $\theta$ an automorphism of a central extension of $L$. In this paper, we study the lattice vertex operator algebra $V_L$ associated with $L$ and its $\theta$-twisted modules. We also discuss the fusion products of $V_L$-modules and completely determine the fusion rules among them.

In this paper, we consider the asymptotic stability for the system of linear delay differential equations. Because of the complicated interactions induced by the delay effects of the system, there are few results of the asymptotic stability for the system of the delay differential equations with multiple delays. Given this fact, we propose the new stability conditions for the system and apply these conditions to some mathematical models for the population dynamics and neural network system described by the system of delay differential equations.