2.
On the unipotent support of character sheaves - Geck, Meinolf; Hézard, David
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).

3.
The restricted Nagata's pairwise algorithm and the Euclidean algorithm - Leu, Ming-Guang
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.

5.
Hyperbolic lengths of some filling geodesics on Riemann surfaces with punctures - Zhang, Chaohui
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
$\pi_{1}(\tilde{S},a)$.

6.
The forcing partial order on a family of braids forced by pseudo-Anosov 3-braids - Kin, Eiko
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.

8.
Equations in $p$-curvature and intertwiners - Tsuchimoto, Yoshifumi
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.

9.
Relative Ext groups, resolutions, and Schanuel classes - Holm, Henrik
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.

10.
Lifespan for radially symmetric solutions to systems of semilinear wave equations with multiple speeds - Katayama, Soichiro
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.

11.
Involutions of compact Riemannian 4-symmetric spaces - Kurihara, Hiroyuki; Tojo, Koji
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 $G$.

12.
Multiplication and composition operators on Lorentz-Bochner spaces - Arora, S.C.; Datt, Gopal; Verma, Satish
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.

13.
Corrected energy of the Reeb distribution of a 3-Sasakian manifold - Perrone, Domenico
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).

14.
Uniqueness for the Brezis-Nirenberg problem on compact Einstein manifolds - Huang, Guangyue; Chen, Wenyi
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.

15.
Wegner estimate and localization for random magnetic fields - Ueki, Naomasa
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.