Let $F$ be a non-Archimedean local field or a finite field.
Let $n$ be a natural number and $k$ be $1$ or $2$.
Consider $G:=\operatorname{GL}_{n+k}(F)$ and let
$M:=\operatorname{GL}_n(F) \times \operatorname{GL}_k(F)\lt G$ be a maximal Levi subgroup.
Let $U\lt G$ be the corresponding unipotent subgroup and let $P=MU$ be the corresponding parabolic subgroup.
Let $J:=J_M^G: \mathcal{M}(G) \to \mathcal{M}(M)$ be the Jacquet functor, i.e., the functor of coinvariants with respect to $U$.
In this paper we prove that $J$ is a multiplicity free functor, i.e.,
$\dim \operatorname{Hom}_M(J(\pi),\rho)\leq 1$,
for any irreducible representations $\pi$ of $G$ and $\rho$ of $M$.
We adapt the classical method of Gelfand and Kazhdan, which proves the ``multiplicity free" property of certain representations to prove the ``multiplicity free" property of certain functors.
At the end we discuss whether other Jacquet functors are multiplicity free.

We show a pointwise estimate for the Fourier
transform on the line involving the number of times the function
changes monotonicity. The contrapositive of the theorem may be used to
find a lower bound to the number of local maxima of a function. We
also show two applications of the theorem. The first is the two weight
problem for the Fourier transform, and the second is estimating the
number of roots of the derivative of a function.

We give precise conditions under which the composition
of a norm with a convex function yields a
uniformly convex function on a Banach space.
Various applications are given to functions of power type.
The results are dualized to study uniform smoothness
and several examples are provided.

We extend results proved by the second author (Amer. J. Math., 2009)
for nonnegatively curved Alexandrov spaces
to general compact Alexandrov spaces $X$ with curvature bounded
below.
The gradient flow of a geodesically convex functional on the quadratic Wasserstein
space $(\mathcal P(X),W_2)$ satisfies the evolution variational inequality.
Moreover, the gradient flow enjoys uniqueness and contractivity.
These results are obtained by proving a first variation formula for
the Wasserstein distance.

In this paper we discuss the existence of mild and classical solutions for a class of abstract non-autonomous
neutral functional differential equations. An application to partial neutral differential equations is considered.

We prove the existence of an approximation function for holomorphic
solutions of a system of real analytic equations. For this we use
ultraproducts and Weierstrass systems introduced by J. Denef and L.
Lipshitz. We also prove a version of the Płoski smoothing theorem in
this case.

We show that any Lipschitz projection-valued function
$p$ on a connected closed Riemannian manifold
can be approximated uniformly by smooth
projection-valued functions $q$ with Lipschitz constant
close to that of $p$.
This answers a question of Rieffel.

We extend the notion of Zindler curve from the Euclidean plane to
normed planes. A characterization of Zindler curves for general
normed planes is given, and the relation between Zindler curves and
curves of constant area-halving distances in such planes is
discussed.

We provide a criterion for the central norm to be
any value in the simple continued fraction expansion of $\sqrt{D}$
for any non-square integer $D>1$. We also provide a simple criterion
for the solvability of the Pell equation $x^2-Dy^2=-1$ in terms of
congruence conditions modulo $D$.

In this paper we define lower, upper, and symmetric completeness and
discuss closure of the sets in product and direct sums. In particular,
we introduce suitable bases for these topologies, which leads us to
investigate completeness of the direct sum and its components. Some
results obtained about $X$-topologies and polars of the neighborhoods.

Let $X$ be a smooth complex projective variety, and let $H \in
\operatorname{Pic}(X)$
be an ample line bundle. Assume that $X$ is covered by rational
curves with degree one with respect to $H$ and with anticanonical
degree greater than or equal to $(\dim X -1)/2$. We prove that there
is a covering family of such curves whose numerical class spans an
extremal ray in the cone of curves $\operatorname{NE}(X)$.

We establish a mixed norm estimate for the Radon transform in
$\mathbb{R}^2$ when the set of directions has fractional dimension.
This estimate is used to prove a result about an exceptional set of directions connected with projections of planar sets. That leads to
a conjecture analogous to a well-known conjecture of Furstenberg.

The study carried out in this paper about some new examples of
Banach spaces, consisting of certain valued fields extensions, is
a typical non-archimedean feature. We determine whether these
extensions are of countable type, have $t$-orthogonal bases, or are
reflexive.
As an application we construct, for a class of base fields, a norm
$\|\cdot\|$ on $c_0$, equivalent to the canonical supremum norm,
without non-zero vectors that are $\|\cdot\|$-orthogonal and such
that there is a multiplication on $c_0$ making $(c_0,\|\cdot\|)$
into a valued field.

Let $(X,\mathcal{B},m,\tau)$ be a dynamical system with $(X,\mathcal{B},m)$ a probability
space and $\tau$ an invertible, measure preserving transformation.
This paper deals with the almost everywhere convergence in $\textrm{L}^1(X)$ of a
sequence of operators of weighted averages. Almost everywhere convergence follows
once we obtain an appropriate maximal estimate and once we provide
a dense class where convergence holds almost everywhere.
The weights are given by convolution products of members of a sequence of probability
measures $\{\nu_i\}$ defined on $\mathbb{Z}$.
We then exhibit cases of such averages where convergence fails.

Frey and Jarden asked if
any abelian variety over a number field $K$
has the infinite Mordell-Weil rank over
the maximal abelian extension $K^{\operatorname{ab}}$.
In this paper,
we give an affirmative answer to their conjecture
for the Jacobian variety
of any smooth projective curve $C$
over $K$
such that $\sharp C(K^{\operatorname{ab}})=\infty$
and for any abelian variety of $\operatorname{GL}_2$-type with trivial character.

We obtain nontrivial estimates of quadratic character sums of division polynomials $\Psi_n(P)$, $n=1,2, \dots$, evaluated at a given point $P$ on an elliptic curve over a finite field of $q$ elements. Our bounds are nontrivial if the order of $P$ is at least $q^{1/2 + \varepsilon}$ for some fixed $\varepsilon > 0$. This work is motivated by an open question about statistical indistinguishability of some cryptographically relevant sequences that was recently brought up by K. Lauter and the second author.

We show that the Schrödinger equation is a lift of Newton's third law
of motion $\nabla^\mathcal W_{\dot \mu} \dot \mu = -\nabla^\mathcal W F(\mu)$ on
the space of probability measures, where derivatives are taken
with respect to the Wasserstein Riemannian metric. Here the potential
$\mu \to F(\mu)$ is the sum of the total classical potential energy $\langle V,\mu\rangle$
of the extended system
and its Fisher information
$ \frac {\hbar^2} 8 \int |\nabla \ln \mu |^2
\,d\mu$. The precise relation is established via a well-known
(Madelung) transform which is shown to be a symplectic submersion
of the standard symplectic
structure of complex valued functions into the
canonical symplectic space over the Wasserstein space.
All computations are conducted in the framework of Otto's formal
Riemannian calculus for optimal transportation of probability
measures.

In this paper we study left invariant Einstein-Randers metrics on compact Lie
groups. First, we give a method to construct left invariant non-Riemannian Einstein-Randers metrics
on a compact Lie group, using the Zermelo navigation data.
Then we prove that this gives a complete classification of left invariant Einstein-Randers metrics on compact simple
Lie groups with the underlying Riemannian metric naturally reductive.
Further, we completely determine the identity component of the group of
isometries for this type of metrics on simple groups. Finally, we study some
geometric properties of such metrics. In particular, we give the formulae of geodesics and flag curvature
of such metrics.

$L_p$ stability and exponential stability are two important concepts
for nonlinear dynamic systems. In this paper, we prove that a
nonlinear exponentially bounded Lipschitzian semigroup is
exponentially stable if and only if the semigroup is $L_p$ stable
for some $p>0$. Based on the equivalence, we derive two sufficient
conditions for exponential stability of the nonlinear semigroup. The
results obtained extend and improve some existing ones.