Inspired by a work of Hislop and Klopp, we prove precise Wegner estimates for three classes of Schrödinger operators, including Pauli Hamiltonians, with random magnetic fields. The support of the site vector potentials may be noncompact (long-range type random perturbation) and, for one class of the operators, the random vector potentials may be unbounded. In particular Gaussian random fields are also treated. Wegner estimates with correct volume dependence are applied to show Hölder estimates of the densities of states. We give also upper bounds on the infimum of the spectrum to show the existence of the Anderson localization near the infimum.

We consider the positive solution of the following semi-linear elliptic equation on the compact Einstein manifolds $M^{n}$ with positive scalar curvature $R_{0}$ \begin{equation*} \Delta_{0}u-\lambda u+f(u)u^{(n+2)/(n-2)}=0, \end{equation*} where $\Delta_{0}$ is the Laplace-Beltrami operator on $M^{n}$. We prove that for $0<\lambda\leq (n-2)R_{0}/(4(n-1))$ and $f'(u)\leq 0$, and at least one of two inequalities is strict, the only positive solution to the above equation is constant. The method here is intrinsic.

In this paper we show that the Reeb distribution on a spherical space form which admits a 3-Sasakian structure minimizes the corrected energy. Analogously for the characteristic distribution of the normal complex contact structure on the complex projective space $\mathbb{C}P^{2m+1}$ induced via the Hopf fibration $S^{1}\hookrightarrow S^{4m+3}\to \mathbb{C}P^{2m+1}$. This last result is a consequence of a more general result on the corrected energy of the characteristic distribution of a compact twistor space over a quaternionic-Kähler manifold with positive scalar curvature (equipped with a normal complex contact metric structure).

In this paper we study the multiplication and composition operators induced by operator valued maps on Bochner spaces (Lorentz-Bochner and rearrangement invariant-Bochner) and discuss their closedness, compactness and spectrum.

Let $G/H$ be a compact 4-symmetric space of inner type such that the dimension of the center $Z(H)$ of $H$ is at most one. In this paper we shall classify involutions of $G$ preserving $H$ for the case where $\dim Z(H)=0$, or $H$ is a centralizer of a toral subgroup of $G$.

We consider the Cauchy problem for a system of semilinear wave equations with multiple propagation speeds in three space dimensions. We obtain the sharp lower bound for the lifespan of radially symmetric solutions to a class of these systems. We also show global existence of radially symmetric solutions to another class of systems with small initial data.

Given a precovering (also called contravariantly finite) class $\mathsf{F}$ there are three natural approaches to a homological dimension with respect to $\mathsf{F}$: One based on Ext functors relative to $\mathsf{F}$, one based on $\mathsf{F}$-resolutions, and one based on Schanuel classes relative to $\mathsf{F}$. In general these approaches do not give the same result. In this paper we study relations between the three approaches above, and we give necessary and sufficient conditions for them to agree.

The equations in $p$-curvatures, which is a key to prove a stable equivalence of Jacobian problem and Dixmier conjecture in the author's previous paper, is provided an easier proof, related to the existence of `intertwining operator'. In an appendix, we show that every symplectic morphism between nonsingular symplectic varieties are of Jacobian 1, regardless of the characteristics.

We prove a rigidity theorem for closed hypersurfaces with constant mean curvature in a symmetric Riemannian manifold, which is a generalization of main results in [3] and [15].

Li-York theorem tells us that a period 3 orbit for a continuous map of the interval into itself implies the existence of a periodic orbit of every period. This paper concerns an analogue of the theorem for homeomorphisms of the 2-dimensional disk. In this case a periodic orbit is specified by a braid type and on the set of all braid types Boyland's dynamical partial order can be defined. We describe the partial order on a family of braids and show that a period 3 orbit of pseudo-Anosov braid type implies the Smale-horseshoe map which is a factor possessing complicated chaotic dynamics.

Let $\tilde{S}$ be a Riemann surface of type $(p,n)$ with $3p-3+n>0$ and $n\geq 1$. In this paper, we give a quantitative common lower bound for the hyperbolic lengths of all filling geodesics on $\tilde{S}$ generated by two parabolic elements in the fundamental group $\pi_{1}(\tilde{S},a)$.

Pencils of curves of minimal genus and slope are determined for smooth surfaces of degree at least seven in the projective 3-space.

In 1971, Samuel generalized Motzkin's idea to give a characterization of Euclidean rings. In this article we will show, from Motzkin and Samuel's point of view, that the concept of the restricted Nagata's pairwise algorithm should exist in the world of mathematics much earlier than the Euclidean algorithm.

Let $G$ be a connected reductive group over $\mathbb{F}_{q}$, where $q$ is large enough and the center of $G$ is connected. We are concerned with Lusztig's theory of character sheaves, a geometric version of the classical character theory of the finite group $G(\mathbb{F}_{q})$. We show that under a certain technical condition, the restriction of a character sheaf to its unipotent support (as defined by Lusztig) is either zero or an irreducible local system. As an application, the generalized Gelfand-Graev characters are shown to form a $\mathbb{Z}$-basis of the $\mathbb{Z}$-module of unipotently supported virtual characters of $G(\mathbb{F}_{q})$ (Kawanaka's conjecture).

In this note, we shall show that Chow-stability and Hilbert-stability in GIT asymptotically coincide. The proof in [5] is simplified in the present form, while a quick review is in [6].