In this paper we study the Cauchy problem for viscous shallow water equations. We work in the Sobolev spaces of index $s>2$ to obtain local solutions for any initial data, and global solutions for small initial data.

We define the concept of symplectic foliation on a symplectic manifold and provide a method of constructing many examples, by using asymptotically holomorphic techniques.

We study slant submanifolds of $S$-manifolds with the smallest dimension, specially minimal submanifolds and establish some relations between them and anti-invariant submanifolds in $S$-manifolds, similar to those ones proved by B.-Y. Chen for slant surfaces and totally real surfaces in Kaehler manifolds.

We give a weighted version of the Sobolev-Lieb-Thirring inequality for suborthonormal functions. In the proof of our result we use $\varphi$-transform of Frazier-Jawerth.

Cotangent type functions in $\mathbb{R}^n$ are used to construct Cauchy kernels and Green kernels on the conformally flat manifolds $\mathbb{R}^n / \mathbb{Z}^k$ where $1\leq k\leq n$. Basic properties of these kernels are discussed including introducing a Cauchy formula, Green's formula, Cauchy transform, Poisson kernel, Szegö kernel and Bergman kernel for certain types of domains. Singular Cauchy integrals are also introduced as are associated Plemelj projection operators. These in turn are used to study Hardy spaces in this context. Also the analogues of Calderón-Zygmund type operators are introduced in this context, together with singular Clifford holomorphic, or monogenic, kernels defined on sector domains in the context of cylinders. Fundamental differences in the context of the $n$-torus arising from a double singularity for the generalized Cauchy kernel on the torus are also discussed.

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.

In 1959, Freiman demonstrated his famous $3k-4$ Theorem which was to be a cornerstone in inverse additive number theory. This result was soon followed by a $3k-3$ Theorem, proved again by Freiman. This result describes the sets of integers $\mathcal{A}$ such that $| \mathcal{A}+\mathcal{A} | \leq 3 | \mathcal{A} | -3$. In the present paper, we prove a $3k-3$-like Theorem in the context of multiple set addition and describe, for any positive integer $j$, the sets of integers $\mathcal{A}$ such that the inequality $|j \mathcal{A} | \leq j(j+1)(| \mathcal{A} | -1)/2$ holds. Freiman's $3k-3$ Theorem is the special case $j=2$ of our result. This result implies, for example, the best known results on a function related to the Diophantine Frobenius number. Actually, our main theorem follows from a more general result on the border of $j\mathcal{A}$.

We study the problem of existence of surfaces in $\mathbb{R}^3$ parametrized on the sphere ${\mathbb S}^2$ with prescribed mean curvature $H$ in the perturbative case, i.e. for $H=H_0+\varepsilon H_1$, where $H_0$ is a nonzero constant, $H_1$ is a $C^2$ function and $\varepsilon$ is a small perturbation parameter.

We study the tridimensional Navier-Stokes equation when the value of the vertical viscosity is zero, in a critical space (invariant by the scaling). We shall prove local in time existence of the solution, respectively global in time when the initial data is small compared with the horizontal viscosity.

A quantitative version of Krein's Theorem on convex hulls of weak compact sets is proved. Some applications to weakly compactly generated Banach spaces are given.

Let $\mathcal{F}$ be a family of functions meromorphic on the plane domain $D$, all of whose zeros are multiple. Suppose that $f'(z)\ne 1$ for all $f\in \mathcal{F}$ and $z\in D.$ Then if $\mathcal{F}$ is quasinormal on $D$, it is quasinormal of order 1 there.

For each weight $k$ and level $N$ square free and without small prime factors, we prove the existence of primitive forms $f_+$ and $f_-$ of weight $k$ and level $N$ such that $$ L(1,\sym^2f_+)\gg_{k}[\log\log(3N)]^{3} $$ and $$ L(1,\sym^2f_-)\ll_{k}[\log\log(3N)]^{-1}. $$ The result comes from a delicate study of the moments of $L(1,\sym^2 f)$. This study gives also results for squarefree levels but with small prime factors. It provides counterexamples to the equivalence between harmonic and natural means.

Let $C^{m , 1} ( \mathbb{R}^n)$ be the space of functions on $\mathbb{R}^n$ whose $m^{\sf th}$ derivatives are Lipschitz 1. For $E \subset \mathbb{R}^n$, let $C^{m , 1} (E)$ be the space of all restrictions to $E$ of functions in $C^{m,1} ( \mathbb{R}^n)$. We show that there exists a bounded linear operator $T: C^{m , 1} (E) \rightarrow C^{m , 1} ( \mathbb{R}^n)$ such that, for any $f \in C^{m , 1} ( E )$, we have $T f = f$ on $E$.