We prove the existence in the sense of sequences of solutions for some integro-differential type equations involving the drift term in the appropriate $H^{2}$ spaces using the fixed point technique when the elliptic problems contain second order differential operators with and without Fredholm property. It is shown that, under the reasonable technical conditions, the convergence in $L^{1}$ of the integral kernels yields the existence and convergence in $H^{2}$ of solutions.

$2$-stratifolds are a generalization of $2$-manifolds in that there are disjoint simple closed branch curves. We obtain a list of all closed $3$-manifolds that have a $2$-stratifold as a spine.

We consider diagrams of links in $S^2$ obtained by projection from $S^3$ with the Hopf map and the minimal crossing number for such diagrams. Knots admitting diagrams with at most one crossing are classified. Some properties of these knots are exhibited. In particular, we establish which of these knots are algebraic and, for such knots, give an answer to a problem posed by Fiedler in [3].

A generalization of the Einstein equations with the cosmological constant is considered for complex line elements. Several second order semilinear partial differential equations are derived from them as semilinear field equations in homogeneous and isotropic spaces. The nonrelativistic limits of the field equations are also considered. The properties of spatial expansion and contraction are studied based on energy estimates of the field equations. Several dissipative and anti-dissipative properties are remarked.

For a graph $G$, the graph cubeahedron $\square_G$ and the graph associahedron $\triangle_G$ are simple convex polytopes which admit (real) toric manifolds. In this paper, we introduce a graph invariant, called the $b$-number, and show that the $b$-numbers compute the Betti numbers of the real toric manifold $X^\mathbb{R}(\square_G)$ corresponding to $\square_G$. The $b$-number is a counterpart of the notion of $a$-number, introduced by S. Choi and the second named author, which computes the Betti numbers of the real toric manifold $X^\mathbb{R}(\triangle_G)$ corresponding to $\triangle_G$. We also study various relationships between $a$-numbers and $b$-numbers from the viewpoint of toric topology. Interestingly, for a forest $G$ and its line graph $L(G)$, the real toric manifolds $X^\mathbb{R}(\triangle_G)$ and $X^\mathbb{R}(\square_{L(G)})$ have the same Betti numbers.

We consider generalized Hodge-Laplace operators $\alpha d \delta + \beta \delta d$ for $\alpha, \beta > 0$ on $p$-forms on compact Riemannian manifolds. In the case of flat tori and round spheres of different radii, we explicitly calculate the spectrum of these operators. Furthermore, we investigate under which circumstances they are isospectral.

We study asymptotic error distributions associated with standard approximation scheme for one-dimensional stochastic differential equations driven by fractional Brownian motions. This problem was studied by, for instance, Gradinaru-Nourdin [6], Neuenkirch and Nourdin [14] and the second named author [13]. The aim of this paper is to extend their results to the case where the equations contain drift terms and simplify the proof of estimates of the remainder terms in [13]. To this end, we represent the approximation solution as the solution of the equation which is obtained by replacing the fractional Brownian path with a perturbed path. We obtain the asymptotic error distribution as a directional derivative of the solution by using this expression.

We introduce a family of quasidistances in ${\mathbb R}^d$, such that some of them are equivalent to natural distances on Carnot groups. We find the sufficient conditions for the balls w.r.t. a quasidistance from our family to be comparable to ellipsoids. Using comparability to ellipsoids we find asymptotics of surface measure of intersections of small balls with linear submanifolds and the conditions for finiteness of the integral w.r.t. the surface measure of negative power of the distance. We provide several examples of Carnot groups, where comparability to ellipsoids can be shown for natural distances, and therefore we can study the asymptotics and finitness of the integrals explicitly. We also show an example of a Carnot group, where the comparability to ellipsoids does not hold.

A degeneration of a singular curve on a toric surface, called a tropicalization, was constructed by E. Shustin. He classified the degeneration of 1-cuspidal curves using polyhedral complexes called tropical curves. In this paper, we define a tropical version of a 1-tacnodal curve, that is, a curve having exactly one singular point whose topological type is $A_3$, and by applying the tropicalization method, we classify tropical curves which correspond to 1-tacnodal curves.

Given a rational fibered surface $f:X \to \mathbb{P}^1$ of genus $g$ we prove the inequality $\frac{6n+5}{n+1}-\frac{9n+12}{2g}\le \lambda_f,$ provided that the genus $g$ is sufficiently high with respect to the gonality $2n+3$ of the general fibre.