For a fixed odd prime $p$ and a representation $\varrho$ of the absolute Galois group of $\mathbb{Q}$ into the projective group ${\rm PGL}_2(\mathbb{F}_p)$, we provide the twisted modular curves whose rational points supply the quadratic $\mathbb{Q}$-curves of degree $N$ prime to $p$ that realize $\varrho$ through the Galois action on their $p$-torsion modules. The modular curve to twist is either the fiber product of $X_0(N)$ and $X(p)$ or a certain quotient of Atkin-Lehner type, depending on the value of $N$ mod $p$. For our purposes, a special care must be taken in fixing rational models for these modular curves and in studying their automorphisms. By performing some genus computations, we obtain as a by-product some finiteness results on the number of quadratic $\mathbb{Q}$-curves of a given degree $N$ realizing $\varrho$.

The motivation of this paper is to study some properties of the local times (when it exists) of the hybrids of empirical and partial-sum processes defined by $$ \bar{A}_n(t)=\sum_{1\leq i \leq n} H(X_i)1_{\{X_i\leq t\}} \epsilon_i, \quad - \infty #x003C; t #x003C; \infty , $$ namely by using knowing results on empirical process and Brownian local times.

On the Bergman space of the unit disk, we consider a class of operators which contain sums of finitely many Toeplitz products with harmonic symbols. We give characterizations of when an operator in that class has finite rank or is compact. Our results provide a unified way of treating several known results.

We exhibit several transformations of surfaces $M$ in $\mathbb{R}^4$: a transformation of flat surfaces that gives surfaces with flat normal bundle (semiumbilical surfaces); and its inverse that from a semiumbilical surface obtains a flat surface; then a one-parameter family of transformations $f$ on flat semiumbilical immersed surfaces (FSIS), such that $df(T_pM)$ is totally orthogonal to $T_pM,$ and that give FSIS. This family satisfies a Bianchi type of permutability property.

In this paper we give a unified and improved treatment to finite dimensionality results for subspaces of $L^{p}$ harmonic sections of Riemannian or Hermitian vector bundles over complete manifolds. The geometric conditions on the manifold are subsumed by the assumption that the Morse index of a related Schr#x00F6;dinger operator is finite. Applications of the finiteness theorem to concrete geometric situations are also presented.

We discuss the algebro-geometric initial value problem for the Toda hierarchy with complex-valued initial data and prove unique solvability globally in time for a set of initial (Dirichlet divisor) data of full measure. To this effect we develop a new algorithm for constructing stationary complex-valued algebro-geometric solutions of the Toda hierarchy, which is of independent interest as it solves the inverse algebro-geometric spectral problem for generally non-self-adjoint Jacobi operators, starting from a suitably chosen set of initial divisors of full measure. Combined with an appropriate first-order system of differential equations with respect to time (a substitute for the well-known Dubrovin equations), this yields the construction of global algebro-geometric solutions of the time-dependent Toda hierarchy. The inherent non-self-adjointness of the underlying Lax (i.e., Jacobi) operator associated with complex-valued coefficients for the Toda hierarchy poses a variety of difficulties that, to the best of our knowledge, are successfully overcome here for the first time. Our approach is not confined to the Toda hierarchy but applies generally to $1+1$-dimensional completely integrable discrete soliton equations.

We analyze the limit of the solutions of an elliptic problem when some reaction and potential terms are concentrated in a neighborhood of a portion $\Gamma$ of the boundary and this neighborhood shrinks to $\Gamma$ as a parameter goes to zero. We prove that this family of solutions converges in certain Sobolev spaces and also in the sup norm, to the solution of an elliptic problem where the reaction term and the concentrating potential are transformed into a flux condition and a potential on $\Gamma$.

Let $V$ be a smooth scheme over a field $k$, and let $\{I_n, n\geq 0\}$ be a filtration of sheaves of ideals in $\mathcal{O}_V$, such that $I_0=\mathcal{O}_V$, and $I_s\cdot I_t\subset I_{s+t}$. In such case $\bigoplus I_n$ is called a Rees algebra. A Rees algebra is said to be a differential algebra if, for any two integers $N > n$ and any differential operator $D$ of order $n$, $D(I_N)\subset I_{N-n}$. Any Rees algebra extends to a smallest differential algebra. There are two extensions of Rees algebras of interest in singularity theory: one defined by taking integral closures, and another by extending the algebra to a differential algebra. We study here some relations between these two extensions, with particular emphasis on the behavior of higher order differentials over arbitrary fields.

We study the heat kernel transform on a nilmanifold $M$ of the Heisenberg group. We show that the image of $L^2(M)$ under this transform is a direct sum of weighted Bergman spaces which are related to twisted Bergman and Hermite-Bergman spaces.

Let $L$ be a generator of a semigroup satisfying the Gaussian upper bounds. A new ${\rm BMO}_L$ space associated with $L$ was recently introduced in [Duong, X. T. and Yan, L.: {New function spaces of BMO type, the John-Nirenberg inequality, interpolation and applications}. \textit{Comm. Pure Appl. Math.} {\bf 58} (2005), 1375-1420] and [Duong, X. T. and Yan, L.: {Duality of Hardy and BMO spaces associated with operators with heat kernels bounds}. \textit{J. Amer. Math. Soc.} {\bf 18} (2005), 943-973]. We discuss applications of the new ${\rm BMO}_L$ spaces in the theory of singular integration. For example we obtain ${\rm BMO}_L$ estimates and interpolation results for fractional powers, purely imaginary powers and spectral multipliers of self adjoint operators. We also demonstrate that the space ${\rm BMO}_L$ might coincide with or might be essentially different from the classical BMO space.

We deal with the existence of positive bound state solutions for a class of stationary nonlinear Schr�dinger equations of the form $$ -\varepsilon^2\Delta u + V(x) u = K(x) u^p,\qquad x\in\mathbb{R}^N, $ where $V, K$ are positive continuous functions and $p > 1$ is subcritical, in a framework which may exclude the existence of ground states. Namely, the potential $V$ is allowed to vanish at infinity and the competing function $K$ does not have to be bounded. In the \emph{semi-classical limit}, i.e. for $\varepsilon\sim 0$, we prove the existence of bound state solutions localized around local minimum points of the auxiliary function $\mathcal{A} = V^\theta K^{-\frac{2}{p-1}}$, where $\theta=(p+1)/(p-1)-N/2$. A special attention is devoted to the qualitative properties of these solutions as $\varepsilon$ goes to zero.

We propose a definition of sampling set for the Nevanlinna and Smirnov classes in the disk and show its equivalence with the notion of determination set for the same classes. We also show the relationship with determination sets for related classes of functions and deduce a characterization of Smirnov sampling sets. For Nevanlinna sampling we give general conditions (necessary or sufficient), from which we obtain precise geometric descriptions in several regular cases.

We give the correct proof of Lemma 3.6 of the paper {\it A Parabolic Quasilinear Problem for Linear Growth Functionals} (Rev. Mat. Iberoamericana {\bf 18} (2002), no. 1, 135-185).