We give sufficient conditions for Mosco convergences for the following three cases: symmetric locally uniformly elliptic diffusions, symmetric Lévy processes, and symmetric jump processes in terms of the $L^1(\mathbb{R}^{d};dx)$-local convergence of the (elliptic) coefficients, the characteristic exponents and the jump density functions, respectively. We stress that the global path properties of the corresponding Markov processes such as recurrence/transience, and conservativeness/explosion are not preserved under Mosco convergences and we give several examples where such situations indeed happen.

We find presentations by generators and relations for the equivariant quantum cohomology rings of the maximal isotropic Grassmannians of types B, C and D, and we find polynomial representatives for the Schubert classes in these rings. These representatives are given in terms of the same Pfaffian formulas which appear in the theory of factorial $P$- and $Q$-Schur functions. After specializing to equivariant cohomology, we interpret the resulting presentations and Pfaffian formulas in terms of Chern classes of tautological bundles.

We study the homogeneous Boltzmann equation with the fractional Laplacian term. Working on the Fourier side we solve the resulting integral equation, and improve a previous result by Y.-K. Cho. We replace the initial data space with a certain space $\mathcal{M}^{\alpha}$ introduced by Morimoto, Wang, and Yang. This space precisely captures the Fourier image of probability measures with bounded fractional moments, providing a more natural initial condition. We show existence of a unique global solution, in addition to the expected maximal growth estimates and stability estimates. As a consequence we obtain a continuous density solution of the original equation.

The famous Russell hypersurface is a smooth complex affine threefold which is diffeomorphic to a euclidean space but not algebraically isomorphic to the three dimensional affine space. This fact was first established by Makar-Limanov, using algebraic minded techniques. In this article, we give an elementary argument which adds a greater insight to the geometry behind the original proof and which also may be applicable in other situations.

We compute the E-polynomials of the moduli spaces of representations of the fundamental group of a complex curve of genus $g=3$ into $\mathrm{SL}(2, \mathbb{C})$, and also of the moduli space of twisted representations. The case of genus $g=1, 2$ has already been done in [12]. We follow the geometric technique introduced in [12], based on stratifying the space of representations, and on the analysis of the behaviour of the E-polynomial under fibrations.

In this paper we show as main results two structure theorems of a compact homogeneous locally conformally Kähler (or shortly l.c.K.) manifold, a holomorphic structure theorem asserting that it has a structure of holomorphic principal fiber bundle over a flag manifold with fiber a $1$-dimensional complex torus, and a metric structure theorem asserting that it is necessarily of Vaisman type. We also discuss and determine l.c.K. reductive Lie groups and compact locally homogeneous l.c.K. manifolds of reductive Lie groups.

We show that for a sufficiently simple surface $S$, if a right-angled Artin group $A(\Gamma)$ embeds into $\mathrm{Mod}(S)$ then $\Gamma$ embeds into the curve graph $\mathcal{C}(S)$ as an induced subgraph. When $S$ is sufficiently complicated, there exists an embedding $A(\Gamma) \to \mathrm{Mod}(S)$ such that $\Gamma$ is not contained in $\mathcal{C}(S)$ as an induced subgraph.

We prove new results about exponential decay rates associated with the two dimensional Schrödinger equation with critical nonlinearity and localized damping. Our article improve incomplete previous results established in [4].

In this paper, we introduce a subposet $\mathcal{L}_{\pi}(G)$ of a poset $\mathcal{N}_{\pi}(G)$ of all non-trivial nilpotent $\pi$-subgroups of a finite group $G$. We examine basic properties of subgroups in $\mathcal{L}_{\pi}(G)$ which contain the notion of both radical $p$-subgroups and centric $p$-subgroups of $G$. It is shown that $\mathcal{L}_{\pi}(G)$ is homotopy equivalent to $\mathcal{N}_{\pi}(G)$. As examples, we investigate in detail the case where symmetric groups.

Chow stability is one notion of Mumford's geometric invariant theory for studying the moduli space of polarized varieties. Kapranov, Sturmfels and Zelevinsky detected that Chow stability of polarized toric varieties is determined by its inherent secondary polytope, which is a polytope whose vertices correspond to regular triangulations of the associated polytope [7]. In this paper, we give a purely convex-geometrical proof that the Chow form of a projective toric variety is $H$-semistable if and only if it is $H$-polystable with respect to the standard complex torus action $H$. This essentially means that Chow semistability is equivalent to Chow polystability for any (not-necessaliry-smooth) projective toric varieties.

For a knot $K$ with $\Delta_{K}(t)\doteq t^{2}-3t+1$ in a homology $3$-sphere, let $M$ be the result of $2/q$-surgery on $K$. We show that an appropriate assumption on the Reidemeister torsion of the universal abelian covering of $M$ implies $q=\pm 1$, if $M$ is a Seifert fibered space.

For the ring of $p$-adic integers, $p$ being a fixed prime, any sequence which plays a similar role to Weyl's irrational rotation has not been proposed yet. We will see that a modified $p$-adic van der Corput sequence provides us with a reasonable counterpart of Weyl's irrational rotation in the ring. We will present a similar random Weyl sampling on the ring to the one proposed by Sugita and Takanobu. In the process of establishing the counterpart, a sampling method based on a function with naturally extended domain to the field of $p$-adic numbers in terms of the additive characters will be mentioned.

We investigate the large-time behavior of the radially symmetric solution for Burgers equation on the exterior of a small ball in multi-dimensional space, where the boundary data and the data at the far field are prescribed. In a previous paper [1], we showed that, for the case in which the boundary data is equal to $0$ or negative, the asymptotic stability is the same as that for the viscous conservation law. In the present paper, it is proved that if the boundary data is positive, the asymptotic state is a superposition of the stationary wave and the rarefaction wave, which is a new wave phenomenon. The proof is given using a standard $L^{2}$ energy method and the characteristic curve method.

In the present paper, we deform isolated singularities of $f\bar{g}$, where $f$ and $g$ are $2$-variable weighted homogeneous complex polynomials, and show that there exists a deformation of $f\bar{g}$ which has only indefinite fold singularities and mixed Morse singularities.

We show a homotopy decomposition of the $p$-localized suspension $\Sigma M_{(p)}$ of a quasitoric manifold $M$ by constructing power maps. As an application we investigate the $p$-localized suspension of the projection $\pi$ from the moment-angle complex onto $M$, from which we deduce its triviality for $p>\dim M/2$. We also discuss non-triviality of $\pi_{(p)}$ and $\Sigma^{\infty}\pi$.

In this paper, we are concerned with the solvability of the initial boundary value problem of a system which describes double-diffusive convection phenomena in some porous medium under general domains, especially unbounded domains. In previous works where the boundedness of the space domain is imposed, some global solvability results have been already derived. However, when we consider our problem in general domains, some compactness theorems are not available. Hence it becomes difficult to follow the same strategies as before. Nevertheless, we can assure the global existence of a unique solution via the contraction method. Moreover, it is revealed that the global solvability holds for higher space dimension and larger class of the initial data than those assumed in previous works.