Let $X(n),$ for $n\in\mathbb{N},$ be the set of all subsets of a metric
space $(X,d)$ of cardinality at most $n.$ The set $X(n)$ equipped
with the Hausdorff metric is called a finite subset space. In
this paper we are concerned with the existence of Lipschitz retractions
$r\colon X(n)\to X(n-1)$ for $n\ge2.$ It is known that such retractions
do not exist if $X$ is the one-dimensional sphere. On the other
hand L. Kovalev has recently established their existence in case $X$
is a Hilbert space and he also posed a question as to whether
or not such Lipschitz retractions exist for $X$ being a Hadamard
space. In the present paper we answer this question in the positive.

Suppose that $G$ is a finite group and $k$ is a field of characteristic
$p\gt 0$. A ghost map is a map in the stable category of
finitely generated $kG$-modules which induces the zero map
in Tate cohomology in all degrees. In an earlier paper we showed
that the
thick subcategory generated by the trivial module
has no nonzero ghost maps if and only if
the Sylow $p$-subgroup of $G$ is cyclic of order 2 or 3.
In this paper we introduce and study variations of ghost
maps.
In particular, we consider the behavior of ghost maps under
restriction
and induction functors. We find all groups satisfying a strong
form
of Freyd's generating hypothesis and show that ghosts can
be detected on a finite range of degrees of Tate cohomology.
We also
consider maps which mimic ghosts in high degrees.

In this note, we study the recurrence and topologically multiple
recurrence of a sequence of operators on Banach spaces.
In particular, we give a sufficient and necessary condition for
a cosine operator function,
induced by a sequence of operators on the Lebesgue space of a
locally compact group, to be topologically multiply recurrent.

The thickness of a graph $G$ is the minimum number
of planar subgraphs whose union is $G.$ A
$t$-minimal graph is a graph of thickness $t$ which contains
no proper subgraph of thickness $t.$ In this paper, upper and
lower bounds are obtained for the thickness, $t(G\Box H)$, of
the Cartesian
product of two graphs $G$ and $H$, in terms of the thickness
$t(G)$ and $t(H)$.
Furthermore, the thickness of the Cartesian product of two planar
graphs and of a $t$-minimal graph and a planar graph are determined.
By using a new planar decomposition of the complete bipartite
graph $K_{4k,4k},$ the thickness of the Cartesian product of
two complete bipartite graphs $K_{n,n}$ and $K_{n,n}$ is also
given, for $n\neq 4k+1$.

There are several kinds of classification problems for real hypersurfaces
in complex two-plane Grassmannians $G_2({\mathbb C}^{m+2})$.
Among them, Suh classified Hopf hypersurfaces $M$ in $G_2({\mathbb
C}^{m+2})$ with Reeb parallel Ricci tensor in Levi-Civita connection.
In this paper, we introduce the notion of generalized Tanaka-Webster
(in shortly, GTW) Reeb parallel Ricci tensor for Hopf hypersurface
$M$ in $G_2({\mathbb C}^{m+2})$. Next, we give a complete classification
of Hopf hypersurfaces in $G_2({\mathbb C}^{m+2})$ with GTW Reeb
parallel Ricci tensor.

We study the distribution of the discrete spectrum of the Schrödinger
operator perturbed by a fast oscillating decaying potential depending
on a small parameter $h$.

We study the total graph of a finite commutative ring. We calculate
its metric dimension in the case when the Jacobson radical of
the ring is nontrivial and we examine the metric dimension of
the total graph of a product of at most two fields, obtaining
either exact values in some cases or bounds in other, depending
on the number of elements in the respective fields.

We propose a version of the classical Artin
approximation
which allows to perturb the variables of the approximated solution. Namely, it is possible to approximate a formal solution of a
Nash equation by a Nash solution in a
compatible way with a given Nash change of variables.
This result is closely related to the so-called nested Artin
approximation and becomes false in the analytic setting. We provide
local and global versions of this approximation in real and complex
geometry together with an application to the Right-Left equivalence
of Nash maps.

We study some geometric properties related to the set $\Pi_X:=
\{
(x,x^*
)\in\mathsf{S}_X\times \mathsf{S}_{X^*}:x^*
(x
)=1
\}$ obtaining two characterizations of Hilbert spaces
in the category of Banach spaces. We also compute the distance
of a generic element $
(h,k
)\in H\oplus_2 H$ to $\Pi_H$ for $H$ a Hilbert space.

The Osgood-Carathéodory theorem asserts that
conformal mappings between Jordan domains extend to homeomorphisms
between their closures.
For multiply-connected domains on Riemann surfaces, similar results
can be reduced to the simply-connected case, but we find it simpler
to deduce such results using a direct analogue of the Carathéodory
reflection principle.

Let $R$ be an associative ring with identity.
First we prove some results about zero-divisor graphs of reversible
rings. Then we study the zero-divisors of the skew power series
ring $R[[x;\alpha]]$, whenever $R$ is reversible and $\alpha$-compatible. Moreover, we compare the diameter and girth of the zero-divisor
graphs of $\Gamma(R)$, $\Gamma(R[x;\alpha,\delta])$ and $\Gamma(R[[x;\alpha]])$,
when
$R$ is reversible and $(\alpha,\delta)$-compatible.

As an application of the theory of
graph-like Legendrian unfoldings, relations of the hidden structures
of caustics and wave front propagations are revealed.

On a real hypersurface $M$ in a non-flat complex
space form there exist the Levi-Civita and the k-th generalized
Tanaka-Webster connections. The aim of the present paper is to
study three dimensional real hypersurfaces in non-flat complex
space forms, whose Lie derivative of the structure Jacobi operator
with respect to the Levi-Civita connections coincides with the
Lie derivative of it with respect to the k-th generalized Tanaka-Webster
connection. The Lie derivatives are considered in direction of
the structure vector field and in directions of any vecro field
orthogonal to the structure vector field.

We show that the conjectural criterion of $p$-incompressibility
for products of projective homogeneous varieties in terms of
the factors, previously known in a few special cases only, holds
in general.
Actually, the proof goes through for a wider class of varieties
which includes the norm varieties associated to symbols in Galois
cohomology of arbitrary degree.

In this paper, using Calderón's
reproducing formula and almost orthogonality estimates, we
prove the lifting property and the embedding theorem of the Triebel-Lizorkin
and Besov spaces associated with Zygmund dilations.

We prove that for the linear scalar delay differential
equation
$$ \dot{x}(t) = - a(t)x(t) + b(t)x(t-1) $$
with non-negative periodic coefficients of period $P\gt 0$, the
stability threshold for the trivial solution is
$r:=\int_{0}^{P}
\left(b(t)-a(t)
\right)\mathrm{d}t=0,$
assuming that $b(t+1)-a(t)$ does not change its sign. By constructing
a class of explicit examples, we show the counter-intuitive result
that in general, $r=0$ is not a stability threshold.

We introduce a weaker notion of (semi)stability for vector bundles
on
reducible curves which does not depend on a choice of polarization,
and
which suffices for many applications of degeneration techniques.
We explore the basic
properties of this alternate notion of (semi)stability. In a
complementary
direction, we record a proof of the existence of semistable extensions
of vector bundles in suitable degenerations.

Let $X$ be smooth projective curve of arbitrary genus $g \gt 3$
over complex numbers. In this short note we will show that the
moduli
space of rank $2$ stable vector bundles with determinant isomorphic
to $L_x$, where $L_x$ denote the line bundle corresponding to
a point $x \in X$ is isomorphic to certain lines in the moduli
space of S-equivalence classes of semistable bundles of rank
2 with
trivial determinant.

The purpose of this paper is to characterize positive measure
$\mu$ on the unit disk such that the analytic
Morrey space $\mathcal{AL}_{p,\eta}$ is boundedly and compactly
embedded to the tent space
$\mathcal{T}_{q,1-\frac{q}{p}(1-\eta)}^{\infty}(\mu)$ for the
case $1\leq q\leq p\lt \infty$
respectively. As an application, these results are used to
establish the boundedness and compactness of integral operators
and multipliers between analytic Morrey spaces.