1.
Oscillations of Hecke eigenvalues at shifted primes - Zhao , Liangyi
In this paper, we are interested in exploring the cancellation of Hecke eigenvalues twisted with an exponential sums whose amplitude is $\sqrt{n}$ at prime arguments.
2.
Multiparameter singular integrals and maximal operators along flat surfaces - Cho
,
Yong-Kum; Hong
,
Sunggeum; Kim
,
Joonil; Yang
,
Chan Woo
We study double Hilbert transforms and maximal functions along
surfaces of the form $(t_1,t_2,\gamma_1(t_1)\gamma_2(t_2))$. The
$L^p(\mathbb{R}^3)$ boundedness of the maximal operator is obtained
if each $\gamma_i$ is a convex increasing and $\gamma_i(0)=0$. The
double Hilbert transform is bounded in $L^p(\mathbb{R}^3)$ if both
$\gamma_i$'s above are extended as even functions. If $\gamma_1$ is
odd, then we need an additional comparability condition on
$\gamma_2$. This result is extended to higher dimensions and the
general hyper-surfaces of the form
$(t_1,\dots,t_{n},\Gamma(t_1,\dots,t_{n}))$ on $\mathbb{R}^{n+1}$.
3.
Extension of $C^{m, \omega}$-Smooth Functions by Linear Operators - Fefferman
,
Charles
Let $C^{m, \omega} ( \mathbb{R}^n)$ be the space of functions on
$\mathbb{R}^n$ whose $m^{\sf th}$ derivatives have modulus of
continuity $\omega$. For $E \subset \mathbb{R}^n$, let $C^{m , \omega}
(E)$ be the space of all restrictions to $E$ of functions in $C^{m ,
\omega} ( \mathbb{R}^n)$. We show that there exists a bounded linear
operator $T: C^{m , \omega} ( E ) \rightarrow C^{m , \omega } (
\mathbb{R}^n)$ such that, for any $f \in C^{m , \omega} ( E )$, we have
$T f = f$ on $E$.
4.
Fitting a $C^m$-Smooth Function to Data II - Fefferman
,
Charles; Klartag
,
Bo'az
We exhibit efficient algorithms to perform the following task: Given
a function $f$ defined on a finite subset $E \subset \mathbb R^n$, compute
a $C^m$ function $F$ on $\mathbb R^n$, with a controlled $C^m$ norm, that
approximates $f$ on the subset $E$.
5.
The $C^m$ Norm of a Function with Prescribed Jets II - Fefferman
,
Charles
We give algorithms to compute a function $F$ on $\mathbb R^n$, having
prescribed Taylor polynomials (or taking prescribed values) at $N$
given points, with the $C^m$-norm of $F$ close to least possible.
6.
Uniformly convex operators and martingale type - Wenzel, Jörg
The concept of uniform convexity of a Banach space was
generalized to linear operators between Banach spaces and
studied by Beauzamy. Under this generalization, a Banach space
$X$ is uniformly convex if and only if its identity map $I_X$
is. Pisier showed that uniformly convex Banach spaces have martingale
type $p$ for some $p>1$. We show that this fact is in general not
true for linear operators. To remedy the situation, we
introduce the new concept of martingale subtype and show,
that it is equivalent, also in the operator case, to the
existence of an equivalent uniformly convex norm on $X$.
In the case of identity maps it is also...
7.
Algebro-Geometric Solutions of the Camassa-Holm hierarchy - Gesztesy, Fritz; Holden, Helge
We provide a detailed treatment of the Camassa-Holm (CH)
hierarchy with special emphasis on its algebro-geometric
solutions. In analogy to other completely integrable hierarchies
of soliton equations such as the KdV or AKNS hierarchies, the CH
hierarchy is recursively constructed by means of a basic
polynomial formalism invoking a spectral parameter. Moreover, we
study Dubrovin-type equations for auxiliary divisors and
associated trace formulas, consider the corresponding
algebro-geometric initial value problem, and derive the theta
function representations of algebro-geometric solutions of the CH
hierarchy.
8.
Small global solutions and the nonrelativistic limit for the nonlinear Dirac equation - Machihara, Shuji; Nakanishi, Kenji; Ozawa, Tohru
In this paper we study the Cauchy problem for the nonlinear Dirac
equation in the Sobolev space $H^s$. We prove the existence and
uniqueness of global solutions for small data in $H^s$ with $s>1$.
The method of proof is based on the Strichartz estimate of $L^2_t$
type for Dirac and Klein-Gordon equations. We also prove that the
solutions of the nonlinear Dirac equation after modulation of phase
converge to the corresponding solutions of the nonlinear
Schröodinger equation as the speed of light tends to infinity.
9.
Lebesgue points for Sobolev functions on metric spaces - Kinnunen, Juha; Latvala, Visa
Our main objective is to study the pointwise behaviour of Sobolev
functions on a metric measure space. We prove that a Sobolev
function has Lebesgue points outside a set of capacity zero if the
measure is doubling. This result seems to be new even for the
weighted Sobolev spaces on Euclidean spaces. The crucial
ingredient of our argument is a maximal function related to
discrete convolution approximations. In particular, we do not use
the Besicovitch covering theorem, extension theorems or
representation formulas for Sobolev functions.
10.
Galois theory of special trinomials - Abhyankar, Shreeram S.
This is the material which I presented at the 60th birthday conference of my
good friend Jos\'{e} Luis Vicente in Seville in September 2001. It is based
on the nine lectures, now called sections, which were given by me at Purdue
in Spring 1997. This should provide a good calculational background for the
Galois theory of vectorial (= additive) polynomials and their iterates.
11.
Integral Closure of Monomial Ideals on Regular Sequences - Kiyek, Karlheinz; Stückrad, Jürgen
It is well known that the integral closure of a monomial
ideal in a polynomial ring in a finite number of indeterminates
over a field is a monomial ideal, again. Let $R$ be a noetherian
ring, and let $(x_1,\ldots,x_d)$ be a regular sequence in $R$
which is contained in the Jacobson radical of $R$.
An ideal $\mathfrak a$ of $R$ is called a monomial ideal with respect
to $(x_1,\ldots,x_d)$ if it can be generated by monomials
$x_1^{i_1}\cdots x_d^{i_d}$. If $x_1R+\cdots + x_dR$ is a radical
ideal of $R$, then we show that the integral closure of a monomial
ideal of $R$ is monomial, again. This result holds, in particular,
for...
12.
Local and Global Theory of the Moduli of Polarized Calabi-Yau Manifolds - Todorov, Andrey
In this paper we review the moduli theory of polarized CY manifolds.
We briefly sketched Kodaira-Spencer-Kuranishi local deformation
theory developed by the author and G. Tian. We also construct the
Teichm\"{u}ller space of polarized CY manifolds following the ideas of I.
R. Shafarevich and I. I. Piatetski-Shapiro. We review the fundamental
result of E. Viehweg about the existence of the course moduli space
of polarized CY manifolds as a quasi-projective variety. Recently S.
Donaldson computed the moment map for the action of the group of
symplectic diffeomorphisms on the space of K\"{a}hler metrics with fixed
class of cohomology. Combining this results with the solution of
Calabi conjecture by Yau one...
13.
Analysis of the free boundary for the $p$-parabolic variational
problem $(p\ge 2)$ - Shahgholian, Henrik
Variational inequalities (free boundaries), governed by the
$p$-parabolic equation ($p\geq 2$), are the objects of
investigation in this paper. Using intrinsic scaling we establish
the behavior of solutions near the free boundary. A consequence
of this is that the time levels of the free boundary are porous
(in $N$-dimension) and therefore its Hausdorff dimension is less
than $N$. In particular the $N$-Lebesgue measure of the free
boundary is zero for each $t$-level.
14.
Critical nonlinear elliptic equations
with singularities and cylindrical symmetry - Badiale, Marino; Serra, Enrico
Motivated by a problem arising in astrophysics we study a
nonlinear elliptic equation in $\mathbb{R}^{N}$ with cylindrical symmetry
and with singularities on a whole subspace of $\mathbb{R}^{N}$. We study
the problem in a variational framework and, as the nonlinearity
also displays a critical behavior, we use some suitable version of
the Concentration-Compactness Principle. We obtain several
results on existence and nonexistence of solutions.
15.
Polynomial growth harmonic functions on complete
Riemannian manifolds - Lee, Yong Hah
In this paper, we give a sharp estimate on the dimension of the
space of polynomial growth harmonic functions with fixed degree on
a complete Riemannian manifold, under various assumptions.
16.
Approximation and symbolic calculus for Toeplitz algebras
on the Bergman space - Suárez, Daniel
If $f\in L^\infty(\mathbb{D})$ let $T_f$ be the Toeplitz operator on
the Bergman space $L^2_a$ of the unit disk $\mathbb{D}$. For a
$C^\ast$-algebra $A\subset L^\infty(\mathbb{D})$ let $\mathfrak{T}(A)$ denote
the closed operator algebra generated by $\{ T_f : f\in A \}$. We
characterize its commutator ideal $\comm(A)$ and the quotient
$\mathfrak{T}(A)/ \mathfrak{C}(A)$ for a wide class of algebras $A$. Also, for
$n\geq 0$ integer, we define the $n$-Berezin transform $B_nS$ of a
bounded operator $S$, and prove that if $f\in L^\infty(\mathbb{D})$ and
$f_n = B_n T_f$ then $T_{f_n} \rightarrow T_f$.
17.
Nonresonant smoothing for coupled wave + transport equations and the Vlasov-Maxwell system - Bouchut, François; Golse, François; Pallard, Christophe
Consider a system consisting of a linear wave equation coupled to
a transport equation:
\begin{equation*}
\Box_{t,x}u =f ,
\end{equation*}
\begin{equation*}
(\partial_t + v(\xi) \cdot \nabla_x)f =P(t,x,\xi, D_\xi)g ,
\end{equation*}
Such a system is called \textit{nonresonant} when the maximum speed
for particles governed by the transport equation is less than the
propagation speed in the wave equation. Velocity averages of
solutions to such nonresonant coupled systems are shown to be more
regular than those of either the wave or the transport equation
alone. This smoothing mechanism is reminiscent of the proof of
existence and uniqueness of $C^1$ solutions of the Vlasov-Maxwell
system by R. Glassey and W. Strauss for time intervals on which
particle momenta remain uniformly...
18.
Resolution of a family of Galois embedding problems - Vela, Montserrat
In this paper we compute the obstruction and the solutions of
cyclic embedding problems given by
$$
(E): \quad 0 \rightarrow \mathbb{Z}/n\mathbb{Z} \rightarrow E \rightarrow \Gamma=\mathbb{Z}/n\mathbb{Z}
\times \stackrel{m)}{\cdots} \times \mathbb{Z}/n\mathbb{Z} \rightarrow 0 ,
$$
with $\mathbb{Z}/n\mathbb{Z}$ trivial $\Gamma$-modulo, finding adequate
representations of $\Gamma$ in the automorphisms group of a
generalized Clifford algebra.
19.
Dyadic BMO on the bidisk - Blasco, Óscar; Pott, Sandra
We give several new characterizations of the dual of the dyadic
Hardy space $H^{1,d}(\mathbb{T}^2)$, the so-called dyadic BMO space
in two variables and denoted ${\mathrm{BMO}}^{\mathit d}_{prod}}$.
These include characterizations in terms of Haar multipliers,
in terms of the ``symmetrised paraproduct'' $\Lambda_b$, in terms
of the rectangular BMO norms of the iterated ``sweeps'', and in
terms of nested commutators with dyadic martingale transforms.
We further explore the connection between ${\mathrm{BMO}}^{\mathit d}_{prod}}$
and John-Nirenberg type inequalities, and study a scale of rectangular
BMO spaces.
20.
Potential Theory for Schröodinger operators
on finite networks - Bendito, Enrique; Carmona, Ángeles; Encinas, Andrés M.
We aim here at analyzing the fundamental properties of positive
semidefinite Schrödinger operators on networks. We show that such
operators correspond to perturbations of the combinatorial Laplacian
through 0-order terms that can be totally negative on a proper subset of
the network. In addition, we prove that these discrete operators have
analogous properties to the ones of elliptic second order operators on
Riemannian manifolds, namely the monotonicity, the minimum principle, the
variational treatment of Dirichlet problems and the condenser principle.
Unlike the continuous case, a discrete Schrödinger operator can be
interpreted as an integral operator and therefore a discrete Potential
Theory with respect to its associated kernel can be built....
1.
