In this paper,we construct an example of three-dimensional complete smooth $\kappa$-noncollapsed manifold, which admits no short time smooth complete and $\kappa$-noncollapsed solutions to the Ricci flow.

In this paper,we prove that a unitary invariant strongly pseudoconvex complex
Finsler metric is a complex Landsberg metric if and if only if it comes from a
unitary invariant Hermitian metric. This implies that there does not exist unitary invariant
complex Landsberg metric unless it comes from a unitary invariant Hermitian
metric.

Some sharp estimates for coefficients, distortion and the growth order are
obtained for harmonic mappings $f \in TL^{\alpha}_H,$ which are locally univalent harmonic mappings
in the unit disk $\mathbb{D}:=\{z:|z|<1\}.$ Moreover, denoting the subclass $TS^{\alpha}_H$ of the
normalized univalent harmonic mappings, we also estimate the growth of $|f|,$ $f \in TS^α_H,$ and their covering theorems.

In this paper, we first show that for every mapping $f$ from a metric space $Ω$ to itself which is continuous off a countable subset of $Ω,$ there exists a nonempty
closed separable subspace $S ⊂ Ω$ so that $f|_S$ is again a self mapping on $S.$ Therefore,
both the fixed point property and the weak fixed point property of a nonempty closed
convex set in a Banach space are separably determined. We then prove that every
separable subspace of $c_0(\Gamma)$ (for any set $\Gamma$) is again lying in $c_0.$ Making use of these
results, we finally presents a simple proof of the famous result: Every non-expansive
self-mapping defined on a nonempty weakly compact convex set of $c_0(\Gamma)$ has a fixed
point.

In this note, for $k$-quasiconformal mappings of a bounded domain into the
complex plane, we give an upper bound of Burkholder integral. Moreover, as an application
we obtain an upper bound of the $L^p$-integral of $\sqrt{J_f}$ and $|Df|$ for certain $K$-quasiconformal mappings.

Let $\mathfrak{F}$ be a non-empty formation of groups, $\tau$ a subgroup functor and $H$ a $p$-subgroup of a finite group $G.$ Let $\overline{G}=G/H_G$ and $\overline{H} =H/H_G.$ We say that $H$ is $\mathfrak{F}_\tau$-$s$-supplemented in $G$ if for some subgroup $\overline{T}$ and some $\tau$-subgroup $\overline{S}$ of $\overline{G}$ contained
in $\overline{H},$ $\overline{H}\overline{T}$ is subnormal in $\overline{G}$ and $\overline{H} ∩ \overline{T} ≤ \overline{S}Z_{\mathfrak{F}}(\overline{G}).$ In this paper, we investigate the
influence of $\mathfrak{F}_\tau$-$s$-supplemented subgroups on the structure of finite groups. Some
new characterizations about solubility of finite groups are obtained.

A class of Finsler metrics with three parameters is constructed. Moreover,
a sufficient and necessary condition for this Finsler metrics to be projectively flat was
obtained.

In 2007, T. Hosokawa and S. Ohno gave the sufficient and necessary conditions
of the boundedness and compactness of differences of composition operators
on the Bloch space. On this base, this paper will generalize these conditions of the
boundedness and compactness of differences of composition operators on the Bloch
type space.

For two simple connected graphs $G_1$ and $G_2$, we introduce a new graph operation
called the total corona $G_1⊛G_2$ on $G_1$ and $G_2$ involving the total graph of $G_1.$ Subsequently, the adjacency (respectively, Laplacian and signless Laplacian) spectra
of $G_1⊛G_2$ are determined in terms of these of a regular graph $G_1$ and an arbitrary
graph $G_2.$ As applications, we construct infinitely many pairs of adjacency (respectively,
Laplacian and signless Laplacian) cospectral graphs. Besides we also compute
the number of spanning trees of $G_1⊛G_2.$

Using an integral identity for a once differentiable mapping, this paper establishes
Hadamard's integral inequalities for $s$-convex and $s$-concave mappings. In
particular, our results improve and extend some known ones in the literature. Finally,
these inequalities are applied to special means.

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