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Using properties of skew-Hamiltonian matrices and classic
connectedness results, we prove that the moduli space
$M_{ort}^0(r,n)$ of stable rank $r$ orthogonal vector bundles
on $\mathbb{P}^2$, with Chern classes $(c_1,c_2)=(0,n)$, and trivial
splitting on the general line, is smooth irreducible of
dimension $(r-2)n-\binom{r}{2}$ for $r=n$ and $n \ge 4$, and
$r=n-1$ and $n\ge 8$. We speculate that the result holds in
greater generality.

We define a free holomorphic function to be a function
that is locally, with respect to the free topology, a bounded
nc-function.
We prove that free holomorphic functions are the functions that
are locally uniformly approximable
by free polynomials. We prove a realization formula and an Oka-Weil
theorem for free analytic functions.

One of the fundamental results in Convex Geometry is Busemann's
theorem, which states that the intersection body of a symmetric convex
body is convex. Thus, it is only natural to ask if there is a
quantitative version of Busemann's theorem, i.e., if the intersection
body operation actually improves convexity. In this paper we
concentrate on the symmetric bodies of revolution to provide several
results on the (strict) improvement of convexity under the
intersection body operation. It is shown that the intersection body of
a symmetric convex body of revolution has the same asymptotic behavior
near the equator as the Euclidean
ball. We apply this result to show that in sufficiently high
dimension the double intersection body of a symmetric convex body of
revolution is very close to an ellipsoid in the Banach-Mazur
distance. We also prove results on the local convexity at the equator
of intersection bodies in the class of star bodies of revolution.

We find, for all sufficiently large $n$ and each $k$, the maximum number of edges in an $n$-vertex graph which does not contain $k+1$ vertex-disjoint triangles.

This extends a result of Moon [Canad. J. Math. 20 (1968), 96-102] which is in turn an extension of Mantel's Theorem. Our result can also be viewed as a density version of the Corrádi-Hajnal Theorem.

We investigate the representation theory of the
crossed-product $C^*$-algebra associated to a compact group $G$
acting on a locally compact space $X$ when the stability subgroups
vary discontinuously.
Our main result applies when $G$ has a principal stability subgroup or
$X$ is locally of finite $G$-orbit type. Then the upper multiplicity
of the representation of the crossed product induced from an
irreducible representation $V$ of a stability subgroup is obtained by
restricting $V$ to a certain closed subgroup of the stability subgroup
and taking the maximum of the multiplicities of the irreducible
summands occurring in the restriction of $V$. As a corollary we obtain
that when the trivial subgroup is a principal stability subgroup, the
crossed product is a Fell algebra if and only if every stability
subgroup is abelian. A second corollary is that the $C^*$-algebra of
the motion group $\mathbb{R}^n\rtimes \operatorname{SO}(n)$ is a Fell algebra. This uses
the classical branching theorem for the special orthogonal group
$\operatorname{SO}(n)$ with respect to $\operatorname{SO}(n-1)$. Since proper transformation
groups are locally induced from the actions of compact groups, we
describe how some of our results can be extended to transformation
groups that are locally proper.

We study bounded derived categories of the category of representations of infinite quivers over a ring $R$. In case $R$ is a commutative noetherian ring with a dualising complex, we investigate an equivalence similar to Grothendieck duality for these categories, while a notion of dualising complex does not apply to them. The quivers we consider are left, resp. right, rooted quivers that are either noetherian or their opposite are noetherian. We also consider reflection functor and generalize a result of Happel to noetherian rings of finite global dimension, instead of fields.

The symmetric group $\mathcal{S}_n$ acts on the power
set $\mathcal{P}(n)$ and also on the set of
square free polynomials in $n$ variables. These
two related representations are analyzed from the stability point
of view. An application is given for the action of the symmetric
group on the cohomology of the pure braid group.

We define a notion of $p$-adic measure on Artin $n$-stacks which are
of strongly finite type over the ring of $p$-adic integers. $p$-adic
measure on schemes can be evaluated by counting points on the
reduction of the scheme modulo $p^n$. We show that an analogous
construction works in the case of Artin stacks as well if we count the
points using the counting measure defined by Toën. As a consequence,
we obtain the result that the Poincaré and Serre series of such
stacks are rational functions, thus extending Denef's result for
varieties. Finally, using motivic integration we show that as $p$
varies, the rationality of the Serre series of an Artin stack defined
over the integers is uniform with respect to $p$.

We study complex projective varieties that parametrize
(finite-dimensional) filiform Lie algebras over ${\mathbb C}$,
using equations derived by Millionshchikov. In the
infinite-dimensional case we concentrate our attention on
${\mathbb N}$-graded Lie algebras of maximal class. As shown by A.
Fialowski
there are only
three isomorphism types of $\mathbb{N}$-graded Lie algebras
$L=\oplus^{\infty}_{i=1} L_i$ of maximal class generated by $L_1$
and $L_2$, $L=\langle L_1, L_2 \rangle$. Vergne described the
structure of these algebras with the property $L=\langle L_1
\rangle$. In this paper we study those generated by the first and
$q$-th components where $q\gt 2$, $L=\langle L_1, L_q \rangle$. Under
some technical condition, there can only be one isomorphism type
of such algebras. For $q=3$ we fully classify them. This gives a
partial answer to a question posed by Millionshchikov.

This paper proves a commutative algebraic extension
of a generalized Skolem-Mahler-Lech theorem due to the first
author.
Let $A$ be a finitely generated commutative $K$-algebra
over a field of characteristic $0$, and let $\sigma$ be
a $K$-algebra automorphism of $A$.
Given ideals $I$ and $J$ of $A$, we show that
the set $S$ of integers $m$ such that
$\sigma^m(I) \supseteq J$ is a finite union of
complete doubly infinite arithmetic progressions in $m$, up to
the addition of a finite set.
Alternatively, this result states that for an affine scheme
$X$ of finite type over $K$,
an automorphism $\sigma \in \operatorname{Aut}_K(X)$, and $Y$ and $Z$
any two closed subschemes of $X$, the set
of integers $m$ with $\sigma^m(Z ) \subseteq Y$ is as above.
The paper presents examples
showing that this result may fail to hold if the affine scheme
$X$ is
not of finite type, or if $X$ is of finite type but the field
$K$ has positive characteristic.

In this article we
study the geometry of the eigenvarieties of unitary groups at points
corresponding to tempered non-stable representations with an
anti-ordinary (a.k.a evil) refinement. We prove that, except in the
case the Galois representation attached to the automorphic form is a
sum of characters, the eigenvariety is non-smooth at such a point,
and that (under some additional hypotheses) its tangent space is big
enough to account for all the relevant Selmer group. We also study the
local reducibility locus
at those points, proving that in general, in contrast with the case of
the eigencurve, it is a proper subscheme of the fiber of the
eigenvariety over the weight space.

We study the hyperspace dynamics induced from generic continuous maps
and from generic homeomorphisms of the Cantor space, with emphasis on the
notions of Li-Yorke chaos, distributional chaos, topological entropy,
chain continuity, shadowing and recurrence.

We investigate the numbers of complex zeros of Littlewood polynomials
$p(z)$ (polynomials with coefficients $\{-1, 1\}$) inside or
on the unit circle $|z|=1$, denoted by $N(p)$ and $U(p)$, respectively.
Two types of Littlewood polynomials are considered: Littlewood
polynomials with one sign change in the sequence of coefficients
and Littlewood polynomials with one negative coefficient. We
obtain explicit formulas for $N(p)$, $U(p)$ for polynomials $p(z)$
of these types. We show that, if $n+1$ is a prime number, then
for each integer $k$, $0 \leq k \leq n-1$, there exists a Littlewood
polynomial $p(z)$ of degree $n$ with $N(p)=k$ and $U(p)=0$. Furthermore,
we describe some cases when the ratios $N(p)/n$ and $U(p)/n$
have limits as $n \to \infty$ and find the corresponding limit
values.

Let ${\bf x}=(x_0,x_1,\ldots)$ be a $N$-periodic sequence of integers
($N\ge1$), and ${\bf s}$ a sturmian sequence with the same barycenter
(and also $N$-periodic, consequently). It is shown that, for affine
functions $\alpha:\mathbb R^\mathbb N_{(N)}\to\mathbb R$ which are increasing relatively
to some order $\le_2$ on $\mathbb R^\mathbb N_{(N)}$ (the space of all $N$-periodic
sequences), the average of $|\alpha|$ on the orbit of ${\bf x}$ is
greater than its average on the orbit of ${\bf s}$.

We provide some new local obstructions to
approximating
tropical curves in
smooth tropical surfaces. These obstructions are based on
a
relation between tropical and complex intersection theories which is
also established here. We give
two applications of the methods developed in this paper.
First we classify all locally irreducible approximable 3-valent fan tropical
curves in a
fan tropical plane.
Secondly, we prove that a generic non-singular
tropical surface
in tropical projective 3-space contains finitely
many approximable tropical lines
if
it is of degree 3, and contains no approximable tropical lines if
it is of degree 4 or more.

Let $w$ be either in the Muckenhoupt class of $A_2(\mathbb{R}^n)$ weights
or in the class of $QC(\mathbb{R}^n)$ weights, and
$L_w:=-w^{-1}\mathop{\mathrm{div}}(A\nabla)$
the degenerate elliptic operator on the Euclidean space $\mathbb{R}^n$,
$n\ge 2$. In this article, the authors establish the non-tangential
maximal function characterization
of the Hardy space $H_{L_w}^p(\mathbb{R}^n)$ associated with $L_w$ for
$p\in (0,1]$ and, when $p\in (\frac{n}{n+1},1]$ and
$w\in A_{q_0}(\mathbb{R}^n)$ with $q_0\in[1,\frac{p(n+1)}n)$,
the authors prove that the associated Riesz transform $\nabla L_w^{-1/2}$
is bounded from $H_{L_w}^p(\mathbb{R}^n)$ to the weighted classical
Hardy space $H_w^p(\mathbb{R}^n)$.

In this paper, we study the stability in the Lyapunov sense of the
equilibrium solutions of discrete or difference Hamiltonian systems
in the plane. First, we perform a detailed study of linear
Hamiltonian systems as a function of the parameters, in particular
we analyze the regular and the degenerate cases. Next, we give a
detailed study of the normal form associated with the linear
Hamiltonian system. At the same time we obtain the conditions under
which we can get stability (in linear approximation) of the
equilibrium solution, classifying all the possible phase diagrams as
a function of the parameters. After that, we study the stability of
the equilibrium solutions of the first order difference system in
the plane associated to mechanical Hamiltonian system and
Hamiltonian system defined by cubic polynomials. Finally, important
differences with the continuous case are pointed out.

We prove two results about nonunital index theory left open in a
previous paper. The
first is that the spectral triple arising from an action of the reals on a $C^*$-algebra with invariant trace
satisfies the hypotheses of the nonunital local index formula. The second result concerns the meaning of spectral flow in the nonunital case. For the special case of paths
arising from the odd
index pairing for smooth spectral triples in the nonunital setting we are able to connect with earlier approaches to the analytic definition of spectral flow.

In this paper, we discuss the isometric embedding problem in
hyperbolic space with nonnegative extrinsic curvature.
We prove a priori bounds for the trace of the second fundamental
form $H$ and extend the result to $n$-dimensions.
We also obtain an estimate for the gradient of the smaller principal
curvature in 2 dimensions.

A representation of the central extension of the
unitary Lie algebra
coordinated with a skew Laurent polynomial ring
is constructed using vertex operators over an integral $\mathbb Z_2$-lattice.
The irreducible decomposition of the representation is explicitly computed and described.
As a by-product, some fundamental representations of affine
Kac-Moody Lie algebra of type $A_n^{(2)}$ are recovered
by the new method.

We obtain results on the unitary equivalence of weak contractions of
class $C_0$ to their Jordan models under an assumption on their
commutants. In particular, our work addresses the case of arbitrary
finite multiplicity. The main tool is the
theory of boundary representations due to Arveson. We also
generalize and improve previously known results concerning unitary
equivalence and similarity to Jordan models when the minimal function
is a Blaschke product.

We study a multimarginal optimal transportation
problem in one dimension. For a symmetric, repulsive cost function, we
show that given a minimizing transport plan, its symmetrization is
induced by a cyclical map, and that the symmetric optimal plan is
unique. The class of costs that we consider includes, in particular,
the Coulomb cost, whose optimal transport problem is strictly related
to the strong interaction limit of Density Functional Theory. In this
last setting, our result justifies some qualitative properties of the
potentials observed in numerical experiments.

Erdős conjectured that for any set $A\subseteq \mathbb{N}$
with positive
lower asymptotic density, there are infinite sets $B,C\subseteq
\mathbb{N}$
such that $B+C\subseteq A$. We verify Erdős' conjecture in
the case that $A$ has Banach density exceeding $\frac{1}{2}$.
As a consequence, we prove that, for $A\subseteq \mathbb{N}$
with
positive Banach density (a much weaker assumption than positive
lower density), we can find infinite $B,C\subseteq \mathbb{N}$
such
that $B+C$ is contained in the union of $A$ and a translate of
$A$. Both of the aforementioned
results are generalized to arbitrary countable
amenable groups. We also provide a positive solution to Erdős'
conjecture for subsets of the natural numbers that are pseudorandom.

An integer is said to be $y$-friable if its greatest prime factor is less than $y$.
In this paper, we obtain estimates for exponential sums
over $y$-friable numbers up to $x$ which are non-trivial
when $y \geq \exp\{c \sqrt{\log x} \log \log x\}$. As a consequence,
we obtain an asymptotic formula for the
number of $y$-friable solutions to the equation $a+b=c$ which is valid
unconditionnally under the same assumption.
We use a contour integration argument based on the saddle point
method, as developped in the context of friable numbers by Hildebrand
and Tenenbaum,
and used by Lagarias, Soundararajan and Harper to study exponential and character sums over friable numbers.

In this article we study exponential trichotomy for infinite dimensional
discrete time dynamical systems. The goal of this article is to prove that
finite time exponential trichotomy conditions allow to derive exponential
trichotomy for any times. We present an application to the case of pseudo
orbits in some neighborhood of a normally hyperbolic set.

Let $\theta\in[0, 1]$ be any irrational number. It is shown that the
extended rotation algebra $\mathcal B_\theta$ introduced in
a previous paper is always an AF algebra.

The algebraic cobordism group of a scheme is generated by cycles that
are proper morphisms from smooth quasiprojective varieties. We prove
that over a field of characteristic zero the quasiprojectivity
assumption can be omitted to get the same theory.

It is proved that if $G=G_1*_{G_3}G_2$ is free product of probability
measure preserving $s$-regular ergodic discrete groupoids amalgamated
over an amenable subgroupoid $G_3$, then the sofic dimension $s(G)$
satisfies the equality
\[
s(G)=\mathfrak{h}(G_1^0)s(G_1)+\mathfrak{h}(G_2^0)s(G_2)-\mathfrak{h}(G_3^0)s(G_3)
\]
where $\mathfrak{h}$ is the normalized Haar measure on $G$.

We found that if $u$ and $v$ are any two unitaries in
a unital $C^*$-algebra with $\|uv-vu\|\lt 2$ and $uvu^*v^*$ commutes with
$u$ and $v,$ then the $C^*$-subalgebra $A_{u,v}$ generated by $u$ and
$v$ is isomorphic to a quotient of some rotation algebra $A_\theta$
provided that $A_{u,v}$ has a unique tracial state.
We also found that the Exel trace formula holds in any unital
$C^*$-algebra.
Let $\theta\in (-1/2, 1/2)$ be a real number. We prove the
following:
For any $\epsilon\gt 0,$ there exists $\delta\gt 0$ satisfying the following:
if $u$ and $v$ are two unitaries in any unital simple $C^*$-algebra
$A$ with tracial rank zero such that
\[
\|uv-e^{2\pi i\theta}vu\|\lt \delta
\text{ and }
{1\over{2\pi i}}\tau(\log(uvu^*v^*))=\theta,
\]
for all tracial state $\tau$ of $A,$ then there exists a pair
of unitaries $\tilde{u}$ and $\tilde{v}$ in $A$
such that
\[
\tilde{u}\tilde{v}=e^{2\pi i\theta} \tilde{v}\tilde{u},\,\,
\|u-\tilde{u}\|\lt \epsilon
\text{ and }
\|v-\tilde{v}\|\lt \epsilon.
\]

Associated with two commutative Banach algebras $A$ and $B$ and
a character $\theta$ of $B$ is a certain Banach algebra product
$A\times_\theta B$, which is a splitting extension of $B$ by
$A$. We investigate two topics for the algebra $A\times_\theta
B$ in relation to the corresponding ones of $A$ and $B$. The
first one is the Bochner-Schoenberg-Eberlein property and the
algebra of Bochner-Schoenberg-Eberlein functions on the spectrum,
whereas the second one concerns the wide range of spectral synthesis
problems for $A\times_\theta B$.

Given a faithful action of a finite group $G$ on an algebraic
curve~$X$ of genus $g_X\geq 2$, we give explicit criteria for
the induced action of~$G$ on the Riemann-Roch space~$H^0(X,\mathcal{O}_X(D))$
to be faithful, where $D$ is a $G$-invariant divisor on $X$ of
degree at least~$2g_X-2$. This leads to a concise answer to the
question when the action of~$G$ on the space~$H^0(X, \Omega_X^{\otimes
m})$ of global holomorphic polydifferentials of order $m$ is
faithful. If $X$ is hyperelliptic, we furthermore provide an
explicit basis of~$H^0(X, \Omega_X^{\otimes m})$. Finally, we
give applications in deformation theory and in coding theory
and we discuss the analogous problem for the action of~$G$ on
the first homology $H_1(X, \mathbb{Z}/m\mathbb{Z})$ if $X$ is a Riemann surface.

Combings of compact, oriented $3$-dimensional manifolds $M$ are
homotopy classes of nowhere vanishing vector fields.
The Euler class of the normal bundle is an invariant of the combing,
and it only depends on the underlying Spin$^c$-structure. A combing
is called torsion
if this Euler class is a torsion element of $H^2(M;\mathbb Z)$. Gompf
introduced a $\mathbb Q$-valued invariant $\theta_G$ of torsion combings
on closed $3$-manifolds, and he showed that $\theta_G$ distinguishes
all torsion combings with the same Spin$^c$-structure.
We give an alternative definition for $\theta_G$ and we express
its variation as a linking number. We define a similar invariant
$p_1$ of combings for manifolds bounded by $S^2$. We relate $p_1$
to the $\Theta$-invariant, which is the simplest configuration
space integral invariant of rational homology $3$-balls, by the
formula $\Theta=\frac14p_1 + 6 \lambda(\hat{M})$ where $\lambda$
is the Casson-Walker invariant.
The article also includes a self-contained presentation of combings
for $3$-manifolds.

Let $p$ be an odd prime. We study the variation of the $p$-rank of
the Selmer group of Abelian varieties with complex multiplication in
certain towers of number fields.

Let $\beta\colon S^{2n+1}\to S^{2n+1}$ be a minimal homeomorphism ($n\ge 1$). We show that
the crossed product $C(S^{2n+1})\rtimes_\beta \mathbb{Z}$ has rational tracial rank at most one.
Let $\Omega$ be a connected compact metric space with finite covering dimension and
with $H^1(\Omega, \mathbb{Z})=\{0\}.$ Suppose that $K_i(C(\Omega))=\mathbb{Z}\oplus G_i,$ where $G_i$ is a finite abelian group, $i=0,1.$
Let $\beta\colon \Omega\to\Omega$ be a minimal homeomorphism. We also show that
$A=C(\Omega)\rtimes_\beta\mathbb{Z}$ has rational tracial rank at most one and is
classifiable.
In particular, this applies to the minimal dynamical systems on
odd dimensional real projective spaces.
This is done by studying minimal homeomorphisms on $X\times \Omega,$ where
$X$ is the Cantor set.

The work of Reid, Chinburg-Hamilton-Long-Reid,
Prasad-Rapinchuk, and the author with Reid have demonstrated that
geodesics or totally geodesic submanifolds can sometimes be used to
determine the commensurability class of an arithmetic manifold. The
main results of this article show that generalizations of these
results to other arithmetic manifolds will require a wide range of
data. Specifically, we prove that certain incommensurable arithmetic
manifolds arising from the semisimple Lie groups of the form
$(\operatorname{SL}(d,\mathbf{R}))^r \times
(\operatorname{SL}(d,\mathbf{C}))^s$ have the same commensurability
classes of totally geodesic submanifolds coming from a fixed
field. This construction is algebraic and shows the failure of
determining, in general, a central simple algebra from subalgebras
over a fixed field. This, in turn, can be viewed in terms of forms of
$\operatorname{SL}_d$ and the failure of determining the form via certain classes of
algebraic subgroups.

We construct one-parameter families of overconvergent Siegel-Hilbert
modular forms. This result has applications to construction of
Galois representations for automorphic forms of non-cohomological
weights.

We consider Tate cycles on an Abelian variety $A$ defined over
a sufficiently large number field $K$ and having complex
multiplication. We show that
there is an effective bound $C = C(A,K)$ so that
to check whether a given cohomology class is a Tate class on
$A$, it suffices to check the action of
Frobenius elements at primes $v$ of norm $ \leq C$.
We also show that for a set of primes $v$ of $K$ of density
$1$, the space of Tate cycles on the special fibre $A_v$ of the
Néron model of $A$ is isomorphic to the space of Tate cycles
on $A$ itself.

In this paper, we give a tropical method for computing Gromov-Witten
type invariants
of Fano manifolds of special type.
This method applies to those Fano manifolds which admit toric
degenerations
to toric Fano varieties with singularities allowing small resolutions.
Examples include (generalized) flag manifolds of type A, and
some moduli space
of rank two bundles on a genus two curve.

We extend the classicial notion of an outer action
$\alpha$ of a group $G$ on a unital ring $A$
to the case when $\alpha$ is a partial action
on ideals, all of which have local units.
We show that if $\alpha$ is an outer partial
action of an abelian group $G$,
then its associated partial skew group
ring $A \star_\alpha G$ is simple if and only if
$A$ is $G$-simple.
This result is applied to partial skew group rings associated with two different types of partial dynamical systems.

This paper is concerned with suitable generalizations of a plane de
Jonquières map to higher dimensional space
$\mathbb{P}^n$ with $n\geq 3$.
For each given point of $\mathbb{P}^n$ there is a subgroup of the entire
Cremona group of dimension $n$
consisting of such maps.
One studies both geometric and group-theoretical properties of this notion.
In the case where $n=3$ one describes an explicit set of generators of
the group and gives a homological characterization
of a basic subgroup thereof.

In this paper we develop a variational method for the Loewner
equation in higher dimensions. As a result we obtain a version of Pontryagin's
maximum principle from optimal control theory for the Loewner equation in
several complex variables. Based on recent work of Arosio, Bracci and
Wold,
we then apply our version of the Pontryagin maximum
principle to obtain first-order necessary conditions for the extremal
mappings for a wide class
of extremal problems over the set of normalized biholomorphic
mappings on the unit ball in $\mathbb{C}^n$.

We study the Mahler measures of certain families of Laurent
polynomials in two and three variables. Each of the known Mahler
measure formulas for these families involves $L$-values of at most one
newform and/or at most one quadratic character. In this paper, we
show, either rigorously or numerically, that the Mahler measures of
some polynomials are related to $L$-values of multiple newforms and
quadratic characters simultaneously. The results suggest that the
number of modular $L$-values appearing in the formulas significantly
depends on the shape of the algebraic value of the parameter chosen
for each polynomial. As a consequence, we also obtain new formulas
relating special values of hypergeometric series evaluated at
algebraic numbers to special values of $L$-functions.

We study the motion of a particle in the hyperbolic plane (embedded in Minkowski space), under the action of a potential that depends only on one variable. This problem is the analogous to the spherical pendulum in a unidirectional force field. However, for the discussion of the hyperbolic plane one has to distinguish three inequivalent cases, depending on the direction of the force field. Symmetry reduction, with respect to groups that are not necessarily compact or even reductive, is carried out by way of Poisson varieties and Hilbert maps. For each case the dynamics is discussed, with special attention to linear potentials.

Let $F$ be a p-adic field of odd residual characteristic. Let
$\overline{GSp_{2n}(F)}$ and $\overline{Sp_{2n}(F)}$ be the metaplectic double covers of the general
symplectic group and the symplectic group attached to the $2n$
dimensional symplectic space over $F$. Let $\sigma$ be a genuine,
possibly reducible, unramified principal series representation of
$\overline{GSp_{2n}(F)}$. In these notes we give an explicit formulas for a spanning
set for the space of Spherical Whittaker functions attached to
$\sigma$. For odd $n$, and generically for even $n$, this spanning set
is a basis. The significant property of this set is that each of its
elements is unchanged under the action of the Weyl group of
$\overline{Sp_{2n}(F)}$.
If $n$ is odd then each element in the set has an equivariant property
that generalizes a uniqueness result of Gelbart, Howe and
Piatetski-Shapiro. Using this symmetric set, we
construct a family of reducible genuine unramified principal series
representations which have more then one generic constituent. This
family contains all the reducible genuine unramified principal series
representations induced from a unitary data and exists only for $n$
even.

In this paper, we study the explicit geography problem of irregular Gorenstein minimal 3-folds of general type. We generalize the classical Noether-Castelnuovo type inequalities for irregular surfaces to irregular 3-folds according to the Albanese dimension.