A quasi-Poisson manifold is a $G$-manifold equipped with an invariant
bivector field whose Schouten bracket is the trivector field generated
by the invariant element in $\wedge^3 \g$ associated to an invariant
inner product. We introduce the concept of the fusion of such
manifolds, and we relate the quasi-Poisson manifolds to the previously
introduced quasi-Hamiltonian manifolds with group-valued moment maps.
We study the symplectic geometry of the moduli spaces
$M_r=M_r(\s^3)$ of closed $n$-gons with fixed side-lengths in the
$3$-sphere. We prove that these moduli spaces have symplectic
structures obtained by reduction of the fusion product of $n$
conjugacy classes in $\SU(2)$ by the diagonal conjugation action of
$\SU(2)$. Here the fusion product of $n$ conjugacy classes is a
Hamiltonian quasi-Poisson $\SU(2)$-manifold in the sense of
\cite{AKSM}. An integrable Hamiltonian system is constructed on
$M_r$ in which the Hamiltonian flows are given by bending polygons
along a maximal collection of nonintersecting diagonals. Finally,
we show the symplectic structure on $M_r$ relates to the
symplectic structure obtained from gauge-theoretic description of
$M_r$. The results of this paper are analogues for the $3$-sphere of
results obtained for $M_r(\h^3)$, the moduli space of $n$-gons with
fixed side-lengths in hyperbolic $3$-space \cite{KMT}, and for
$M_r(\E^3)$, the moduli space of $n$-gons with fixed side-lengths in
$\E^3$ \cite{KM1}.
We verify a generalization of (3.3) from \cite{Le} proving
that the homotopy type of the Milnor fiber of a reduced
hypersurface singularity depends only on the embedded
topological type of the singularity. In particular, using
\cite{Za,Li1,Oh1,Gau} for irreducible quasi-ordinary germs,
it depends only on the normalized distinguished pairs of the
singularity. The main result of the paper provides an explicit
formula for the Euler-characteristic of the Milnor fiber in the
surface case.
Let $b_1,\dots,b_5$ be non-zero integers and $n$ any integer. Suppose
that $b_1 + \cdots + b_5 \equiv n \pmod{24}$ and $(b_i,b_j) = 1$ for
$1 \leq i < j \leq 5$. In this paper we prove that
\begin{enumerate}[(ii)]
\item[(i)] if $b_j$ are not all of the same sign, then the above
quadratic equation has prime solutions satisfying $p_j \ll \sqrt{|n|}
+ \max \{|b_j|\}^{20+\ve}$; and
\item[(ii)] if all $b_j$ are positive and $n \gg \max \{|b_j|\}^{41+
\ve}$, then the quadratic equation $b_1 p_1^2 + \cdots + b_5 p_5^2 =
n$ is soluble in primes $p_j$.
\end{enumerate}
We prepare for a comparison of global trace formulas of general linear
groups and their metaplectic coverings. In particular, we generalize
the local metaplectic correspondence of Flicker and Kazhdan and
describe the terms expected to appear in the invariant trace formulas
of the above covering groups. The conjectural trace formulas are
then placed into a form suitable for comparison.
It is shown that simple stably projectionless $\C^S*$-algebras which
are inductive limits of certain specified building blocks with trivial
$\K$-theory are classified by their cone of positive traces with
distinguished subset. This is the first example of an isomorphism
theorem verifying the conjecture of Elliott for a subclass of the
stably projectionless algebras.
We introduce the concept of logarithmic dimension of a compact set.
In terms of this magnitude, the extension property and the diametral
dimension of spaces $\calE(K)$ can be described for Cantor-type
compact sets.
We describe the set of numbers $\sigma_k(z_1,\ldots,z_{n+1})$, where
$z_1,\ldots,z_{n+1}$ are complex numbers of modulus $1$ for which
$z_1z_2\cdots z_{n+1}=1$, and $\sigma_k$ denotes the $k$-th
elementary symmetric polynomial. Consequently, we give sharp
constraints on the coefficients of a complex polynomial all of whose
roots are of the same modulus. Another application is the calculation
of the spectrum of certain adjacency operators arising naturally
on a building of type ${\tilde A}_n$.
Soit $G$ un groupe réductif connexe défini sur un corps $p$-adique $F$ et $\ggo$
son algèbre de Lie. Les intégrales orbitales pondérées sur $\ggo(F)$ sont des
distributions $J_M(X,f)$---$f$ est une fonction test---indexées par les
sous-groupes de Lévi $M$ de $G$ et les éléments semi-simples réguliers
$X \in \mgo(F)\cap \ggo_{\reg}$. Leurs analogues sur $G$ sont les principales
composantes du côté géométrique des formules des traces locale et globale d'Arthur.
Si $M=G$, on retrouve les intégrales orbitales invariantes qui, vues comme fonction
de $X$, sont bornées sur $\mgo(F)\cap \ggo_{\reg}$~: c'est un résultat bien connu
de Harish-Chandra. Si $M \subsetneq G$, les intégrales orbitales pondérées
explosent au voisinage des éléments singuliers. Nous construisons dans cet article
de nouvelles intégrales orbitales pondérées $J_M^b(X,f)$, égales à $J_M(X,f)$ à
un terme correctif près, qui tout en conservant les principales propriétés des
précédentes (comportement par conjugaison, développement en germes, {\it etc.})
restent bornées quand $X$ parcourt $\mgo(F)\cap\ggo_{\reg}$. Nous montrons
également que les intégrales orbitales pondérées globales, associées à des
éléments semi-simples réguliers, se décomposent en produits de ces nouvelles
intégrales locales.
We study convergence in weighted convolution algebras $L^1(\omega)$ on
$R^+$, with the weights chosen such that the corresponding weighted
space $M(\omega)$ of measures is also a Banach algebra and is the dual
space of a natural related space of continuous functions. We
determine convergence factor $\eta$ for which
weak$^\ast$-convergence of $\{\lambda_n\}$ to $\lambda$ in $M(\omega)$
implies norm convergence of $\lambda_n \ast f$ to $\lambda \ast f$ in
$L^1 (\omega\eta)$. We find necessary and sufficent conditions which
depend on $\omega$ and $f$ and also find necessary and sufficent
conditions for $\eta$ to be a convergence factor for all $L^1(\omega)$
and all $f$ in $L^1(\omega)$. We also give some applications to the
structure of weighted convolution algebras. As a preliminary result
we observe that $\eta$ is a convergence factor for $\omega$ and $f$ if
and only if convolution by $f$ is a compact operator from $M(\omega)$
(or $L^1(\omega)$) to $L^1 (\omega\eta)$.
Let $B$ be the unit ball of $\bb{C}^n$ with respect to an arbitrary norm. We
prove that the analog of the Carath\'eodory set, {\it i.e.} the set of normalized
holomorphic mappings from $B$ into $\bb{C}^n$ of ``positive real part'', is
compact. This leads to improvements in the existence theorems for the Loewner
differential equation in several complex variables. We investigate a subset
of the normalized biholomorphic mappings of $B$ which arises in the study of
the Loewner equation, namely the set $S^0(B)$ of mappings which have
parametric representation. For the case of the unit polydisc these mappings
were studied by Poreda, and on the Euclidean unit ball they were studied by
Kohr. As in Kohr's work, we consider subsets of $S^0(B)$ obtained by placing
restrictions on the mapping from the Carath\'eodory set which occurs in the
Loewner equation. We obtain growth and covering theorems for these subsets of
$S^0(B)$ as well as coefficient estimates, and consider various examples.
Also we shall see that in higher dimensions there exist mappings in $S(B)$
which can be imbedded in Loewner chains, but which do not have parametric
representation.
We study the cohomology of connected components of Shimura varieties
$S_{K^p}$ coming from the group $\GSp_{2g}$, by an approach modeled on
the stabilization of the twisted trace formula, due to Kottwitz and
Shelstad. More precisely, for each character $\olomega$ on
the group of connected components of $S_{K^p}$ we define an operator
$L(\omega)$ on the cohomology groups with compact supports $H^i_c
(S_{K^p}, \olbbQ_\ell)$, and then we prove that the virtual
trace of the composition of $L(\omega)$ with a Hecke operator $f$ away
from $p$ and a sufficiently high power of a geometric Frobenius
$\Phi^r_p$, can be expressed as a sum of $\omega$-{\em weighted}
(twisted) orbital integrals (where $\omega$-{\em weighted} means that
the orbital integrals and twisted orbital integrals occuring here each
have a weighting factor coming from the character $\olomega$).
As the crucial step, we define and study a new invariant $\alpha_1
(\gamma_0; \gamma, \delta)$ which is a refinement of the invariant
$\alpha (\gamma_0; \gamma, \delta)$ defined by Kottwitz. This is done
by using a theorem of Reimann and Zink.
In this paper, we present a smooth framework for some aspects of the
``geometry of CW complexes'', in the sense of Buoncristiano, Rourke
and Sanderson \cite{[BRS]}. We then apply these ideas to Morse
theory, in order to generalize results of Franks \cite{[F]} and
Iriye-Kono \cite{[IK]}.
More precisely, consider a Morse function $f$ on a closed manifold
$M$. We investigate the relations between the attaching maps in a CW
complex determined by $f$, and the moduli spaces of gradient flow
lines of $f$, with respect to some Riemannian metric on~$M$.
We investigate exceptional sets associated with various additive
problems involving sums of cubes. By developing a method wherein an
exponential sum over the set of exceptions is employed explicitly
within the Hardy-Littlewood method, we are better able to exploit
excess variables. By way of illustration, we show that the number of
odd integers not divisible by $9$, and not exceeding $X$, that fail to
have a representation as the sum of $7$ cubes of prime numbers, is
$O(X^{23/36+\eps})$. For sums of eight cubes of prime numbers, the
corresponding number of exceptional integers is $O(X^{11/36+\eps})$.
On d\'emontre un th\'eor\`eme de Vorono\"\i\ (caract\'erisation des
maxima locaux de l'invariant d'Hermite) pour les familles de r\'eseaux
param\'etr\'ees par les espaces sym\'etriques irr\'e\-ductibles non
exceptionnels de type non compact.
We prove a theorem of Vorono\"\i\ type (characterisation of local
maxima of the Hermite invariant) for the lattices parametrized by
irreducible nonexceptional symmetric spaces of noncompact type.
An explicit formula is derived for the logarithmic Mahler measure
$m(P)$ of $P(x,y) = p(x)y - q(x)$, where $p(x)$ and $q(x)$ are
cyclotomic. This is used to find many examples of such polynomials
for which $m(P)$ is rationally related to the Dedekind zeta value
$\zeta_F (2)$ for certain quadratic and quartic fields.
We compute the category of perverse sheaves on Hermitian symmetric
spaces in types~A and D, constructible with respect to the Schubert
stratification. The calculation is microlocal, and uses the action of
the Borel group to study the geometry of the conormal variety
$\Lambda$.
Nous \'etablissons un r\'esultat d'approximation forte pour des
processus bivari\'es ayant une partie gaus\-sien\-ne et une partie
empirique. Ce r\'esultat apporte un nouveau point de vue sur deux
th\'eor\`emes hongrois bidimensionnels \'etablis pr\'ec\'edemment,
concernant l'approximation par un processus de Kiefer d'un
processus empirique uniforme unidimensionnel et l'approximation par
un pont brownien bidimensionnel d'un processus empirique uniforme
bidimensionnel. Nous les enrichissons un peu et montrons que sous leur
nouvelle forme ils ne sont que deux \'enonc\'es d'un m\^eme r\'esultat.
We establish a strong approximation result for bivariate processes
containing a Gaussian part and an empirical part. This result leads
to a new point of view on two Hungarian bidimensional theorems
previously established, about the approximation of an unidimensional
uniform empirical process by a Kiefer process and the approximation of
a bidimensional uniform empirical process by a bidimensional Brownian
bridge. We enrich them slightly and we prove that, under their new
fashion, they are but two statements of the same result.
We characterize embeddability of algebraic varieties into smooth toric
varieties and prevarieties. Our embedding results hold also in an
equivariant context and thus generalize a well-known embedding theorem
of Sumihiro on quasiprojective $G$-varieties. The main idea is to
reduce the embedding problem to the affine case. This is done by
constructing equivariant affine conoids, a tool which extends the
concept of an equivariant affine cone over a projective $G$-variety to
a more general framework.
Given a matrix $A$, let $\mathcal{O}(A)$ denote the orbit of $A$ under a
certain group action such as
\begin{enumerate}[(4)]
\item[(1)] $U(m) \otimes U(n)$ acting on $m \times n$ complex matrices
$A$ by $(U,V)*A = UAV^t$,
\item[(2)] $O(m) \otimes O(n)$ or $\SO(m) \otimes \SO(n)$ acting on $m
\times n$ real matrices $A$ by $(U,V)*A = UAV^t$,
\item[(3)] $U(n)$ acting on $n \times n$ complex symmetric or
skew-symmetric matrices $A$ by $U*A = UAU^t$,
\item[(4)] $O(n)$ or $\SO(n)$ acting on $n \times n$ real symmetric or
skew-symmetric matrices $A$ by $U*A = UAU^t$.
\end{enumerate}
Denote by
$$
\mathcal{O}(A_1,\dots,A_k) = \{X_1 + \cdots + X_k : X_i \in
\mathcal{O}(A_i), i = 1,\dots,k\}
$$
the joint orbit of the matrices $A_1,\dots,A_k$. We study the set of
diagonals or partial diagonals of matrices in $\mathcal{O}(A_1,\dots,A_k)$,
{\it i.e.}, the set of vectors $(d_1,\dots,d_r)$ whose entries lie
in the $(1,j_1),\dots,(r,j_r)$ positions of a matrix in $\mathcal{O}(A_1,
\dots,A_k)$ for some distinct column indices $j_1,\dots,j_r$. In many
cases, complete description of these sets is given in terms of the
inequalities involving the singular values of $A_1,\dots,A_k$. We
also characterize those extreme matrices for which the equality cases
hold. Furthermore, some convexity properties of the joint orbits are
considered. These extend many classical results on matrix
inequalities, and answer some questions by Miranda. Related results
on the joint orbit $\mathcal{O}(A_1,\dots,A_k)$ of complex
Hermitian matrices under the action of unitary similarities are
also discussed.
We extend the basic theory of Lie algebras of affine algebraic groups
to the case of pro-affine algebraic groups over an algebraically
closed field $K$ of characteristic 0. However, some modifications
are needed in some extensions. So we introduce the pro-discrete
topology on the Lie algebra $\mathcal{L}(G)$ of the pro-affine
algebraic group $G$ over $K$, which is discrete in the
finite-dimensional case and linearly compact in general. As an
example, if $L$ is any sub Lie algebra of $\mathcal{L}(G)$, we show
that the closure of $[L,L]$ in $\mathcal{L}(G)$ is algebraic in
$\mathcal{L}(G)$.
We also discuss the Hopf algebra of representative functions $H(L)$ of
a residually finite dimensional Lie algebra $L$. As an example, we
show that if $L$ is a sub Lie algebra of $\mathcal{L}(G)$ and $G$
is connected, then the canonical Hopf algebra morphism from $K[G]$
into $H(L)$ is injective if and only if $L$ is algebraically dense
in $\mathcal{L}(G)$.
We give conditions which determine if $\cat$ of a map go up when
extending over a cofibre. We apply this to reprove a result of
Roitberg giving an example of a CW complex $Z$ such that $\cat(Z)=2$
but every skeleton of $Z$ is of category $1$. We also find conditions
when $\cat (f\times g) < \cat(f) + \cat(g)$. We apply our result to
show that under suitable conditions for rational maps $f$, $\mcat(f) <
\cat(f)$ is equivalent to $\cat(f) = \cat (f\times \id_{S^n})$. Many
examples with $\mcat(f) < \cat(f)$ satisfying our conditions are
constructed. We also answer a question of Iwase by constructing
$p$-local spaces $X$ such that $\cat (X\times S^1) = \cat(X) = 2$. In
fact for our spaces and every $Y \not\simeq *$, $\cat (X\times Y) \leq
\cat(Y) +1 < \cat(Y) + \cat(X)$. We show that this same $X$ has the
property $\cat(X) = \cat (X\times X) = \cl (X\times X) = 2$.
We consider a unitary representation of a discrete countable abelian
group on a separable Hilbert space which is associated to a cyclic
generalized frame multiresolution analysis. We extend Robertson's
theorem to apply to frames generated by the action of the group.
Within this setup we use Stone's theorem and the theory of projection
valued measures to analyze wandering frame collections. This yields a
functional analytic method of constructing a wavelet from a
generalized frame multi\-resolution analysis in terms of the frame
scaling vectors. We then explicitly apply our results to the action
of the integers given by translations on $L^2({\mathbb R})$.
Consider the sixth Painlev\'e equation~(P$_6$) below where $\alpha$,
$\beta$, $\gamma$ and $\delta$ are complex parameters. We prove the
necessary and sufficient conditions for the existence of rational
solutions of equation~(P$_6$) in term of special relations among the
parameters. The number of distinct rational solutions in each case is
exactly one or two or infinite. And each of them may be generated by
means of transformation group found by Okamoto [7] and B\"acklund
transformations found by Fokas and Yortsos [4]. A list of rational
solutions is included in the appendix. For the sake of completeness,
we collected all the corresponding results of other five Painlev\'e
equations (P$_1$)--(P$_5$) below, which have been investigated by many
authors [1]--[7].
We study the local $L$-functions for Levi subgroups in split spinor
groups defined via the Langlands-Shahidi method and prove a conjecture
on their holomorphy in a half plane. These results have been used in
the work of Kim and Shahidi on the functorial product for $\GL_2
\times \GL_3$.
We investigate representations of the Cuntz algebra $\mathcal{O}_2$
on antisymmetric Fock space $F_a (\mathcal{K}_1)$ defined by
isometric implementers of certain quasi-free endomorphisms of the
CAR algebra in pure quasi-free states $\varphi_{P_1}$. We pay
corresponding to these representations and the Fock special
attention to the vector states on $\mathcal{O}_2$ vacuum, for which
we obtain explicit formulae. Restricting these states to the
gauge-invariant subalgebra $\mathcal{F}_2$, we find that for
natural choices of implementers, they are again pure quasi-free and
are, in fact, essentially the states $\varphi_{P_1}$. We proceed to
consider the case for an arbitrary pair of implementers, and deduce
that these Cuntz algebra representations are irreducible, as are their
restrictions to $\mathcal{F}_2$.
The endomorphisms of $B \bigl( F_a (\mathcal{K}_1) \bigr)$ associated
with these representations of $\mathcal{O}_2$ are also considered.
We develop a method for deriving integral representations of certain
orthogonal polynomials as moments. These moment representations are
applied to find linear and multilinear generating functions for
$q$-orthogonal polynomials. As a byproduct we establish new
transformation formulas for combinations of basic hypergeometric
functions, including a new representation of the $q$-exponential
function $\mathcal{E}_q$.
Let $\mathbf{A}$ be a $k$-element algebra whose chief factor size is
$c$. We show that if $\mathbf{B}$ is in the variety generated by
$\mathbf{A}$, then any abelian chief factor of $\mathbf{B}$ that is
not strongly abelian has size at most $c^{k-1}$. This solves
Problem~5 of {\it The Structure of Finite Algebras}, by D.~Hobby and
R.~McKenzie. We refine this bound to $c$ in the situation where the
variety generated by $\mathbf{A}$ omits type $\mathbf{1}$. As a
generalization, we bound the size of multitraces of types~$\mathbf{1}$,
$\mathbf{2}$, and $\mathbf{3}$ by extending coordinatization
theory. Finally, we exhibit some examples of bad behavior, even in
varieties satisfying a congruence identity.
We introduce the notion of strongly projective graph, and characterise
these graphs in terms of their neighbourhood poset. We describe certain
exponential graphs associated to complete graphs and odd cycles. We
extend and generalise a result of Greenwell and Lov\'asz \cite{GreLov}:
if a connected graph $G$ does not admit a homomorphism to $K$, where $K$
is an odd cycle or a complete graph on at least 3 vertices, then the
graph $G \times K^s$ admits, up to automorphisms of $K$, exactly $s$
homomorphisms to $K$.
We study the moderate growth generalized Whittaker functions,
associated to a unitary character $\psi$ of a unipotent subgroup,
for the non-tempered cohomological representation of $G = \Sp
(2,\mathbb{R})$. Through an explicit calculation of a holonomic
system which characterizes these functions we observe that their
existence is determined by the including relation between the real
nilpotent coadjoint $G$-orbit of $\psi$ in
$\mathfrak{g}_{\mathbb{R}}^\ast$ and the asymptotic support of the
cohomological representation.
Willis's structure theory of totally disconnected locally compact groups
is investigated in the context of permutation actions. This leads to new
interpretations of the basic concepts in the theory and also to new proofs
of the fundamental theorems and to several new results. The treatment of
Willis's theory is self-contained and full proofs are given of all the
fundamental results.
Let $\pi$ be an irreducible generalized principal series
representation of $G = \Sp(2,\mathbb{R})$ induced from its Jacobi parabolic
subgroup. We show that the space of algebraic intertwining operators
from $\pi$ to the representation induced from an irreducible
admissible representation of $\SL(2,\mathbb{C})$ in $G$ is at most one
dimensional. Spherical functions in the title are the images of
$K$-finite vectors by this intertwining operator. We obtain an
integral expression of Mellin-Barnes type for the radial part of our
spherical function.
This paper gives a characterization of valuations that follow the
singular infinitely near points of plane vector fields, using the
notion of L'H\^opital valuation, which generalizes a well known classical
condition. With that tool, we give a valuative description of vector
fields with infinite solutions, singularities with rational quotient
of eigenvalues in its linear part, and polynomial vector fields with
transcendental solutions, among other results.
We give optimal upper and lower bounds for the function
$H(x,s)=\sum_{n\geq 1}\frac{1}{(x+n)^s}$ for $x\geq 0$ and $s>1$. These
bounds improve the standard inequalities with integrals. We deduce from them
inequalities about Riemann's $\zeta$ function, and we give a conjecture
about the monotonicity of the function
$s\mapsto[(s-1)\zeta(s)]^{\frac{1}{s-1}}$. Some applications concern the
convexity of functions related to Euler's $\Gamma$ function and optimal
majorization of elementary functions of Baskakov's operators. Then, the
result proved for the function $x\mapsto x^{-s}$ is extended to completely
monotonic functions. This leads to easy evaluation of the order of the
generating series of some arithmetical functions when $z$ tends to 1. The
last part is concerned with the class of non negative decreasing convex
functions on $]0,+\infty[$, integrable at infinity.
Nous prouvons un encadrement optimal pour la quantit\'e
$H(x,s)=\sum_{n\geq 1}\frac{1}{(x+n)^s}$ pour $x\geq 0$ et $s>1$, qui
am\'eliore l'encadrement standard par des int\'egrales. Cet encadrement
entra{\^\i}ne des in\'egalit\'es sur la fonction $\zeta$ de Riemann, et
am\`ene \`a conjecturer la monotonie de la fonction
$s\mapsto[(s-1)\zeta(s)]^{\frac{1}{s-1}}$. On donne des applications \`a
l'\'etude de la convexit\'e de fonctions li\'ees \`a la fonction $\Gamma$
d'Euler et \`a la majoration optimale des fonctions \'el\'ementaires
intervenant dans les op\'erateurs de Baskakov. Puis, nous \'etendons aux
fonctions compl\`etement monotones sur $]0,+\infty[$ les r\'esultats \'etablis
pour la fonction $x\mapsto x^{-s}$, et nous en d\'eduisons des preuves
\'el\'ementaires du comportement, quand $z$ tend vers $1$, des s\'eries
g\'en\'eratrices de certaines fonctions arithm\'etiques. Enfin, nous
prouvons qu'une partie du r\'esultat se g\'en\'eralise \`a une classe de
fonctions convexes positives d\'ecroissantes.
Given a homogeneous elliptic partial differential operator $L$ with constant
complex coefficients and a class of functions (jet-distributions) which
are defined on a (relatively) closed subset of a domain $\Omega$ in $\mathbf{R}^n$ and
which belong locally to a Banach space $V$, we consider the problem of
approximating in the norm of $V$ the functions in this class by ``analytic''
and ``meromorphic'' solutions of the equation $Lu=0$. We establish new Roth,
Arakelyan (including tangential) and Carleman type theorems for a large class
of Banach spaces $V$ and operators $L$. Important applications to boundary
value problems of solutions of homogeneous elliptic partial differential
equations are obtained, including the solution of a generalized Dirichlet
problem.
In this article we state and prove precise theorems on the homotopy
classification of graded categorical groups and their homomorphisms.
The results use equivariant group cohomology, and they are applied to
show a treatment of the general equivariant group extension problem.
We study the resonances of the operator $P(h) = -\Delta_x + V(x) +
\varphi(hx)$. Here $V$ is a periodic potential, $\varphi$ a
decreasing perturbation and $h$ a small positive constant. We prove
the existence of shape resonances near the edges of the spectral bands
of $P_0 = -\Delta_x + V(x)$, and we give its asymptotic expansions in
powers of $h^{\frac12}$.
We investigate the bifurcation of limit cycles in one-parameter
unfoldings of quadractic differential systems in the plane having a
degenerate critical point at infinity. It is shown that there are
three types of quadratic systems possessing an elliptic critical point
which bifurcates from infinity together with eventual limit cycles
around it. We establish that these limit cycles can be studied by
performing a degenerate transformation which brings the system to a
small perturbation of certain well-known reversible systems having a
center. The corresponding displacement function is then expanded in a
Puiseux series with respect to the small parameter and its
coefficients are expressed in terms of Abelian integrals. Finally, we
investigate in more detail four of the cases, among them the elliptic
case (Bogdanov-Takens system) and the isochronous center
$\mathcal{S}_3$. We show that in each of these cases the
corresponding vector space of bifurcation functions has the Chebishev
property: the number of the zeros of each function is less than the
dimension of the vector space. To prove this we construct the
bifurcation diagram of zeros of certain Abelian integrals in a complex
domain.
We consider the Cauchy problem for the cubic nonlinear Schr\"odinger
equation in one space dimension
\begin{equation}
\begin{cases}
iu_t + \frac12 u_{xx} + \bar{u}^3 = 0,
& \text{$t \in \mathbf{R}$, $x \in \mathbf{R}$,} \\
u(0,x) = u_0(x), & \text{$x \in \mathbf{R}$.}
\end{cases}
\label{A}
\end{equation}
Cubic type nonlinearities in one space dimension heuristically appear
to be critical for large time. We study the global existence and
large time asymptotic behavior of solutions to the Cauchy problem
(\ref{A}). We prove that if the initial data $u_0 \in
\mathbf{H}^{1,0} \cap \mathbf{H}^{0,1}$ are small and such that
$\sup_{|\xi|\leq 1} |\arg \mathcal{F} u_0 (\xi) - \frac{\pi n}{2}|
< \frac{\pi}{8}$ for some $n \in \mathbf{Z}$, and $\inf_{|\xi|\leq
1} |\mathcal{F} u_0 (\xi)| >0$, then the solution has an additional
logarithmic time-decay in the short range region $|x| \leq
\sqrt{t}$. In the far region $|x| > \sqrt{t}$ the asymptotics have
a quasi-linear character.
We present a concise explicit expression for the heat trace
coefficients of spheres. Our formulas yield certain combinatorial
identities which are proved following ideas of D.~Zeilberger. In
particular, these identities allow to recover in a surprising way
some known formulas for the heat trace asymptotics. Our approach is
based on a method for computation of heat invariants developed in [P].
In this paper, we investigate projectivity in the category of operator
spaces. In particular, we show that the Fourier algebra of a locally
compact group $G$ is operator biprojective if and only if $G$ is
discrete.
By means of the Pucci operator, we construct a function $u_0$, which plays
an essential role in our considerations, and give the existence and regularity
theorems for the bounded viscosity solutions of the generalized Dirichlet
problems of second order fully nonlinear elliptic equations on the general
bounded domains, which may be irregular. The approximation method, the accretive
operator technique and the Caffarelli's perturbation theory are used.
Form domains are characterized for regular $2n$-th order differential
equations subject to general self-adjoint boundary conditions
depending affinely on the eigenparameter. Corresponding modes of
convergence for eigenfunction expansions are studied, including
uniform convergence of the first $n-1$ derivatives.
Let $X$ be a complex Banach space and let $B_p(X)$ denote the
vector-valued Bergman space on the unit disc for $1\le p<\infty$. A
sequence $(T_n)_n$ of bounded operators between two Banach spaces $X$
and $Y$ defines a multiplier between $B_p(X)$ and $B_q(Y)$
(resp.\ $B_p(X)$ and $\ell_q(Y)$) if for any function $f(z) =
\sum_{n=0}^\infty x_n z^n$ in $B_p(X)$ we have that $g(z) =
\sum_{n=0}^\infty T_n (x_n) z^n$ belongs to $B_q(Y)$ (resp.\
$\bigl( T_n (x_n) \bigr)_n \in \ell_q(Y)$). Several results on these
multipliers are obtained, some of them depending upon the Fourier or
Rademacher type of the spaces $X$ and $Y$. New properties defined by
the vector-valued version of certain inequalities for Taylor
coefficients of functions in $B_p(X)$ are introduced.
Let $X\colon\mathbb{R}^2\to\mathbb{R}^2$ be a $C^1$ map. Denote by $\Spec(X)$ the set of
(complex) eigenvalues of $\DX_p$ when $p$ varies in $\mathbb{R}^2$. If there exists
$\epsilon >0$ such that $\Spec(X)\cap(-\epsilon,\epsilon)=\emptyset$, then
$X$ is injective. Some applications of this result to the real Keller Jacobian
conjecture are discussed.
Generically, one can attach to a $\mathbf{Q}$-curve $C$ octahedral representations
$\rho\colon\Gal(\bar{\mathbf{Q}}/\mathbf{Q})\rightarrow\GL_2(\bar\mathbf{F}_3)$
coming from the Galois action on the $3$-torsion of those abelian varieties of
$\GL_2$-type whose building block is $C$. When $C$ is defined over a quadratic
field and has an isogeny of degree $2$ to its Galois conjugate, there exist
such representations $\rho$ having image into $\GL_2(\mathbf{F}_9)$. Going
the other way, we can ask which $\mod 3$ octahedral representations $\rho$ of
$\Gal(\bar\mathbf{Q}/\mathbf{Q})$ arise from $\mathbf{Q}$-curves in the above
sense. We characterize those arising from quadratic $\mathbf{Q}$-curves of
degree $2$. The approach makes use of Galois embedding techniques in
$\GL_2(\mathbf{F}_9)$, and the characterization can be given in terms of a
quartic polynomial defining the $\mathcal{S}_4$-extension of $\mathbf{Q}$
corresponding to the projective representation $\bar{\rho}$.
Let $\mathbf{R}/R$ be a quadratic extension of finite, commutative,
local and principal rings of odd characteristic. Denote by
$\mathbf{U}_n (\mathbf{R})$ the unitary group of rank $n$ associated
to $\mathbf{R}/R$. The Weil representation of $\mathbf{U}_n
(\mathbf{R})$ is defined and its character is explicitly computed.
We classify all connected $n$-dimensional complex manifolds admitting
effective actions of the unitary group $U_n$ by biholomorphic
transformations. One consequence of this classification is a
characterization of $\CC^n$ by its automorphism group.
We regard a system of left invariant vector fields $\mathcal{X}=\{X_1,\dots,X_k\}$
satisfying the H\"ormander condition and the related Carnot-Carath\'eodory metric on a
unimodular Lie group $G$. We define Besov spaces corresponding to the sub-Laplacian
$\Delta=\sum X_i^2$ both with positive and negative smoothness. The atomic
decomposition of the spaces is given. In consequence we get the distributional
characterization of the Hausdorff dimension of Borel subsets with the Haar measure
zero.
Continued fractions associated with $\GL_3 (\mathbf{Z})$ are
introduced and applied to find fundamental units in a two-parameter
family of complex cubic fields.
Let $X$ be a separated finite type scheme over a noetherian base ring
$\mathbb{K}$. There is a complex $\widehat{\mathcal{C}}^{\cdot} (X)$
of topological $\mathcal{O}_X$-modules, called the complete Hochschild
chain complex of $X$. To any $\mathcal{O}_X$-module
$\mathcal{M}$---not necessarily quasi-coherent---we assign the complex
$\mathcal{H}om^{\cont}_{\mathcal{O}_X} \bigl(
\widehat{\mathcal{C}}^{\cdot} (X), \mathcal{M} \bigr)$ of continuous
Hochschild cochains with values in $\mathcal{M}$. Our first main
result is that when $X$ is smooth over $\mathbb{K}$ there is a
functorial isomorphism
$$
\mathcal{H}om^{\cont}_{\mathcal{O}_X} \bigl(
\widehat{\mathcal{C}}^{\cdot} (X), \mathcal{M} \bigr) \cong \R
\mathcal{H}om_{\mathcal{O}_{X^2}} (\mathcal{O}_X, \mathcal{M})
$$
in the derived category $\mathsf{D} (\Mod \mathcal{O}_{X^2})$, where
$X^2 := X \times_{\mathbb{K}} X$.
The second main result is that if $X$ is smooth of relative dimension
$n$ and $n!$ is invertible in $\mathbb{K}$, then the standard maps
$\pi \colon \widehat{\mathcal{C}}^{-q} (X) \to \Omega^q_{X/
\mathbb{K}}$ induce a quasi-isomorphism
$$
\mathcal{H}om_{\mathcal{O}_X} \Bigl( \bigoplus_q \Omega^q_{X/
\mathbb{K}} [q], \mathcal{M} \Bigr) \to
\mathcal{H}om^{\cont}_{\mathcal{O}_X} \bigl(
\widehat{\mathcal{C}}^{\cdot} (X), \mathcal{M} \bigr).
$$
When $\mathcal{M} = \mathcal{O}_X$ this is the quasi-isomorphism
underlying the Kontsevich Formality Theorem.
Combining the two results above we deduce a decomposition of the
global Hochschild cohomology
$$
\Ext^i_{\mathcal{O}_{X^2}} (\mathcal{O}_X, \mathcal{M}) \cong
\bigoplus_q \H^{i-q} \Bigl( X, \bigl( \bigwedge^q_{\mathcal{O}_X}
\mathcal{T}_{X/\mathbb{K}} \bigr) \otimes_{\mathcal{O}_X} \mathcal{M}
\Bigr),
$$
where $\mathcal{T}_{X/\mathbb{K}}$ is the relative tangent sheaf.