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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....

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