Let $\Omega = (\omega_{1}, \dots, \omega_{n-k})$ be differential $1$-forms with polynomial coefficients in ${\bf R}^{n}$. A Pfaffian manifold of $\Omega$ is by definition a maximal integral $k$-manifold of $\Omega$. It is shown that the number of homeomorphism classes of all Pfaffian manifolds of Rolle Type of $\Omega$ is finite and, moreover, bounded by a computable function in variables $n$, $k$ and the degree of $\omega_{1}, \dots, \omega_{n-k}$. Finiteness is proved also in any o-minimal structure.

We give also an example of a semi-algebraic $C^{1}$ differential form on a semialgebraic $C^{2}$ $3$-manifold whose Pfaffian manifolds have homeomorphism classes of the cardinality of continuum. Hence the cardinality of all manifolds is the continuum (not countable).

In this paper we shall establish some generalization of Nevanlinna's Lemma on Logarithmic Derivative to the case of meromorphic maps from a finite analytic covering space over the $m$-dimensional complex affine space ${\mathbb C}^{m}$ to a smooth complex projective variety. Then we shall apply this to "the Second Main Theorem" in Nevanlinna theory in several complex variables.

The existence and uniqueness of eigenvalues and positive eigenfunctions for some quasilinear elliptic systems are considered. Some necessary and sufficient conditions which guarantee the existence and uniqueness of eigenvalues and positive eigenfunctions are given.

For a real valued periodic smooth function $u$ on $\R$, $n \ge 0$, one defines the osculating polynomial $\varphi_{s}$ (of order $2n+1$at a point $s \in \R$ to be the unique trigonometric polynomial of degree $n$, whose value and first $2n$ derivatives at $s$ coincide with those of $u$ at $s$. We will say that a point $s$ is a clean maximal flex (resp. clean minimal flex) of the function $u$ on $S^{1}$ if and only if $\varphi_{s} \ge u$ (resp.\ $\varphi_{s} \le u$) and the preimage $(\varphi-u)^{-1}(0)$ is connected. We prove that any smooth periodic function $u$ has any smooth periodic function $u$ has at least $n+1$ clean maximal flexes of order $2n+1$ and at least $n+1$ clean minimal flexes of order $2n+1$. The assertion is clearly reminiscent of Morse theory and generalizes the classical four vertex theorem for convex plane curves.

Let $g(f)$ be a Littlewood-Paley square function of $f$, which belongs to Campanato spaces $L^{p,\alpha}(1 < p <\infty,-{n\over p} \leq\alpha <1)$. We prove that if $g(f)(x_0)$ exists (i.e. $g(f)(x_0)<\infty$) for a single point $x_0 \in R^n$, then $g(f)(x)$ exists almost everywhere in $R^n$ and $\|g(f)\|_{L^{p,\alpha}}\leq C\|f\|_{L^{p,\alpha}}$.Thus we give an improvement of some earlier results such as in [8], where it is always needed to assume $g(f)(x)$ exists in a set of positive measure in order to get the a.e. existence and boundedness of $g(f)(x)$.

In this paper we give certain linear relations between the Fourier coefficients of Siegel modular forms that are obtained from Ikeda lifts.

We construct a discriminant-preserving map from the set of orbits in the space of quadruples of quinary alternating forms over the integers to the set of isomorphism classes of quintic rings. This map may be regarded as an analogue of the famous map from the set of equivalence classes of integral binary cubic forms to the set of isomorphism classes of cubic rings and may be expected to have similar applications. We show that the ring of integers of every quintic number field lies in the image of the map. These results have been used to establish an upper bound on the number of quintic number fields with bounded discriminant.

In this paper, the commutators of singular integrals with rough kernels are considered. By the method of block decomposition for kernel function and Fourier transform estimates, some new results about the $L^p(\mathbb{R}^n)$ boundedness for these commutators are obtained.