Let $( X,g) $ be a complete noncompact Kähler manifold, of
dimension $n\geq2,$ with positive Ricci curvature and of standard type
(see the definition below). N. Mok proved that $X$ can be
compactified, i.e., $X$ is biholomorphic to a quasi-projective
variety$.$ The aim of this paper is to prove that the $L^{2}$
holomorphic sections of the line bundle $K_{X}^{-q}$ and the volume
form of the metric $g$ have no essential singularities near the
divisor at infinity. As a consequence we obtain a comparison between
the volume forms of the Kähler metric $g$ and of the Fubini--Study
metric induced on $X$. In the case of $\dim_{\mathbb{C} }X=2,$ we
establish a relation between the number of components of the divisor
$D$ and the dimension of the groups $H^{i}( \overline{X},
\Omega_{\overline{X}}^{1}( \log D) )$.

In this paper we consider an elliptic system with an inverse square potential and
critical Sobolev exponent in a bounded domain of $\mathbb{R}^N$. By variational
methods we study the existence results.

We decompose the restriction of ramified principal series
representations of the $p$-adic group $\mathrm{GL}(3,\mathrm{k})$ to its
maximal compact subgroup $K=\mathrm{GL}(3,R)$. Its decomposition is
dependent on the degree of ramification of the inducing characters and
can be characterized in terms of filtrations of the Iwahori subgroup
in $K$. We establish several irreducibility results and illustrate
the decomposition with some examples.

In this paper, we introduce a new algebraic notion, weakly symmetric
Lie algebras, to give an algebraic description of an
interesting class of homogeneous Riemann--Finsler spaces, weakly symmetric
Finsler spaces. Using this new definition, we are able to give a
classification of weakly symmetric Finsler spaces with dimensions $2$
and $3$. Finally, we show that all the non-Riemannian reversible weakly
symmetric Finsler spaces we find are non-Berwaldian and with vanishing
S-curvature. This means that reversible non-Berwaldian Finsler spaces
with vanishing S-curvature may exist at large. Hence the generalized
volume comparison theorems due to Z. Shen are valid for a rather large
class of Finsler spaces.

We present the explicit formulas for the projectors on the generalized
eigenspaces associated with some eigenvalues for linear neutral functional
differential equations (NFDE) in $L^{p}$ spaces by using integrated
semigroup theory. The analysis is based on the main result
established elsewhere by the authors and results by Magal and Ruan
on non-densely defined Cauchy problem.
We formulate the NFDE as a non-densely defined Cauchy problem and obtain
some spectral properties from which we then derive explicit formulas for
the projectors on the generalized eigenspaces associated with some
eigenvalues. Such explicit formulas are important in studying bifurcations
in some semi-linear problems.

Let $G$ be a connected semisimple split group over a $p$-adic field.
We establish the explicit link between principal nilpotent
orbits and the irreducible constituents of principal series
in terms of $L$-group objects.

Let $A$ and $B$ be $n\times n$ complex Hermitian (or real symmetric) matrices
with eigenvalues $a_1 \ge \dots \ge a_n$ and $b_1 \ge \dots \ge b_n$.
All possible inertia values, ranks, and multiple eigenvalues
of $A + B$ are determined. Extension of the results to the sum of $k$ matrices
with $k > 2$ and connections of the results to other subjects such
as algebraic combinatorics are also discussed.

In the finite von Neumann algebra setting, we introduce the concept
of a perturbation determinant associated with a pair of self-adjoint
elements $H_0$ and $H$ in the algebra and relate it to the concept of
the de la Harpe--Skandalis homotopy invariant determinant associated
with piecewise $C^1$-paths of operators joining $H_0$ and $H$. We
obtain an analog of Krein's formula that relates the perturbation
determinant and the spectral shift function and, based on this
relation, we derive subsequently (i) the Birman--Solomyak formula for
a general non-linear perturbation, (ii) a universality of a spectral
averaging, and (iii) a generalization of the
Dixmier--Fuglede--Kadison differentiation formula.

Let $H$ be the Hilbert class field of a CM number field $K$ with
maximal totally real subfield $F$ of degree $n$ over $\mathbb{Q}$. We
evaluate the second term in the Taylor expansion at $s=0$ of the
Galois-equivariant $L$-function $\Theta_{S_{\infty}}(s)$ associated to
the unramified abelian characters of $\operatorname{Gal}(H/K)$. This is an identity
in the group ring $\mathbb{C}[\operatorname{Gal}(H/K)]$ expressing
$\Theta^{(n)}_{S_{\infty}}(0)$ as essentially a linear combination of
logarithms of special values $\{\Psi(z_{\sigma})\}$, where $\Psi\colon
\mathbb{H}^{n} \rightarrow \mathbb{R}$ is a Hilbert modular function for a congruence
subgroup of $SL_{2}(\mathcal{O}_{F})$ and $\{z_{\sigma}: \sigma \in
\operatorname{Gal}(H/K)\}$ are CM points on a universal Hilbert modular variety. We
apply this result to express the relative class number $h_{H}/h_{K}$
as a rational multiple of the determinant of an $(h_{K}-1) \times
(h_{K}-1)$ matrix of logarithms of ratios of special values
$\Psi(z_{\sigma})$, thus giving rise to candidates for higher analogs
of elliptic units. Finally, we obtain a product formula for
$\Psi(z_{\sigma})$ in terms of exponentials of special values of
$L$-functions.

A new decomposition, the mutually aposyndetic decomposition of
homogeneous continua into closed, homogeneous sets is introduced. This
decomposition is respected by homeomorphisms and topologically
unique. Its quotient is a mutually aposyndetic homogeneous continuum,
and in all known examples, as well as in some general cases, the
members of the decomposition are semi-indecomposable continua. As
applications, we show that hereditarily decomposable homogeneous
continua and path connected homogeneous continua are mutually
aposyndetic. A class of new examples of homogeneous continua is
defined. The mutually aposyndetic decomposition of each of these
continua is non-trivial and different from Jones' aposyndetic
decomposition.

We introduce a wide subclass ${\mathcal F}(X,\omega)$ of
quasi-plurisubharmonic functions in a compact Kähler manifold, on
which the complex Monge-Ampère operator is well defined and the
convergence theorem is valid. We also prove that ${\mathcal F}(X,\omega)$
is a convex cone and includes all quasi-plurisubharmonic functions
that are in the Cegrell class.