We study the generic properties of symplectic relations. Local models of symplectic relations are described and the corresponding local symplectic invariants are derived. A stratification of the Lagrangian Grassmannian in the product symplectic space $(N\x M,\pi _M^*\w _M-\pi _N^*\w _N)$ is constructed and global homological properties of the strata are investigated.

We study low regularity solutions for a class of nonlinear wave equations. We prove local existence for large data in both the subcritical and critical cases and global existence for small data in the critical case. We apply our results to the study of the Dirac-Klein-Gordon system with generalized Yukawa interaction.

Let $T$ be a semi-hyponormal operator on a complex separable Hilbert space. In this paper, we give Helton-Howe type trace formulae of smooth functions associated with the polar decomposition $T = U|T|$ and improve Theorem 10 of [5] by this result.

We study the theory of scattering for the Wave-Schr\"odinger system with Yukawa type coupling in space dimension 3. We prove in particular the existence of modified wave operators for that system with no size restriction on the wave data in the framework of a direct method which requires smallness of the Schr\"odinger data, and we determine the asymptotic behaviour in time of solutions in the range of the wave operators.

We show that an Osserman-type inequality holds for spacelike surfaces of constant mean curvature 1 with singularities and with elliptic ends in de Sitter 3-space. An immersed end of a constant mean curvature 1 surface is an "elliptic end" if the monodromy representation at the end is diagonalizable with eigenvalues in the unit circle. We also give a necessary and sufficient condition for equality in the inequality to hold, and in the process of doing this we derive a condition for determining when elliptic ends are embedded.

Three-dimensional rotating Navier-Stokes equations are considered with a constant Coriolis parameter $\Omega$ and initial data nondecreasing at infinity. In contrast to the non-rotating case ($\Omega=0$), it is shown for the problem with rotation ($\Omega \neq 0$) that Green's function corresponding to the linear problem (Stokes + Coriolis combined operator) does not belong to $L^1({\mathbb R}^3)$. Moreover, the corresponding integral operator is unbounded in the space $L^{\infty}_{\sigma}({\mathbb R}^3)$ of solenoidal vector fields in ${\mathbb R}^3$ and the linear (Stokes+Coriolis) combined operator does not generate a semigroup in $L^{\infty}_{\sigma}({\mathbb R}^3)$. Local in time unique solvability of the rotating Navier-Stokes equations is proven for initial velocity fields in the space $L^{\infty}_{\sigma,a}({\mathbb R}^3)$ which consists of $L^{\infty}$ solenoidal vector fields satisfying vertical averaging property such that their baroclinic component belongs to a homogeneous Besov space ${\dot B}_{\infty,1}^0$ which is smaller than $L^\infty$ but still contains various periodic and almost periodic functions. This restriction of initial data to $L^{\infty}_{\sigma,a}({\mathbb R}^3)$ which is a subspace of $L^{\infty}_{\sigma}({\mathbb R}^3)$ is essential for the combined linear operator (Stokes + Coriolis) to generate a semigroup. Using the rotation transformation, we also obtain local in time solvability of the classical 3D Navier-Stokes equations in ${\mathbb R}^3$ with initial velocity and vorticity of the form $\mbox{\bf{V}}(0)=\tilde{\mbox{\bf{V}}}_0(y) + \frac{\Omega}{2} e_3 \times y$, $\mbox{curl} \mbox{\bf{V}}(0)=\mbox{curl} \tilde{\mbox{\bf{V}}}_0(y) + \Omega e_3$ where $\tilde{\mbox{\bf{V}}}_0(y) \in L^{\infty}_{\sigma,a}({\mathbb R}^3)$.

We study the spectral gaps of the Schr{\"o}dinger operators $$H_{1}=-\frac{d^{2}}{dx^{2}}+\sum^{\infty}_{l=-\infty}( \beta_{1}\delta^{\prime}(x-\kappa-2\pi l)+\beta_{2}\delta^{\prime}(x-2\pi l))\quad {\rm in}\quad L^{2}({\mathbb R}),$$ $$H_{2}=-\frac{d^{2}}{dx^{2}}+\sum^{\infty}_{l=-\infty}( \beta_{1}\delta(x-\kappa-2\pi l)+\beta_{2}\delta(x-2\pi l))\quad {\rm in}\quad L^{2}({\mathbb R}),$$ where $\kappa\in (0,2\pi)$ and $\beta_{1},\beta_{2}\in{\mathbb R}\backslash\{0\}$ are parameters. Given $j\in{\mathbb N}$, we determine whether the $j$th gap of $H_{k}$ is absent or not for $k=1,2$.

Let $\cA$ be a tensor category and let $\cV$ denote the category of vector spaces. A distributor on $\cA$ is a functor $\cA^{\op}\times \cA\to \cV$. We are concerned with distributors with two-sided $\cA$-action. Those distributors form a tensor category, which we denoted by ${}_{\cA}\bD(\cA,\cA)_{\cA}$. The functor category $\Hom(\cA^{\op},\cV)$ is also a tensor category and has the center $\bZ(\Hom(\cA^{\op},\cV))$. We show that if $\cA$ is rigid, then ${}_{\cA}\bD(\cA,\cA)_{\cA}$ and $\bZ(\Hom(\cA^{\op},\cV))$ are equivalent as tensor categories.

Every torsion abelian group has basic subgroups and all its basic subgroups are isomorphic. We extend the concept of basic subgroups from torsion abelian groups to arbitrary abelian groups. The generalized basic subgroups are called mixed basic subgroups. An example of a mixed group is given in which not all mixed basic subgroups are isomorphic. This example also shows that a mixed basic subgroup of a splitting group need not be splitting.

Let $S$ be a blocking semioval in an arbitrary projective plane $\Pi$ of order 9 which meets some line in 8 points. According to Dover in $[2]$, $20\leq \vert S\vert\leq 24$. In $[8]$ one of the authors showed that if $\Pi$ is desarguesian, then $22\leq\vert S\vert\leq 24$. In this note all blocking semiovals with this property in all non-desarguesian projective planes of order 9 are completely determined. In any non-desarguesian plane $\Pi$ it is shown that $21\leq \vert S\vert \leq 24$ and for each $i\in \{ 21,22,23,24\}$ there exist blocking semiovals of size $i$ which meet some line in 8 points. Therefore, the Dover's bound is not sharp.

A numerical method is proposed and investigated for the hypersingular integral equations defined in Banach spaces. The hypersingular integral equations belong to a wider class of singular integral equations having much more stronger singularities. The proposed approximation method is an extension beyond the quadrature method.Moreover an error estimates theory is introduced for the hypersingular integral equations by proving the proper theorem. Finally, the inequalities valid between the exact solutions of the hypersingular integral equations and the corresponding approximate solutions, are proposed and proved.

Let $\Bbb K$ be an algebraically closed field and let $X\subset \Bbb K^l$ be an $n-$dimensional affine variety of degree $D.$ We give a sharp estimation of the degree of the set of non-properness for generically-finite separable and dominant mapping $f=(f_1,...,f_n): X\to \Bbb K^n$. We show that such a mapping must be finite, provided it has a sufficiently large geometric degree. Moreover, we estimate the \L ojasiewicz exponent at infinity of a polynomial mapping $f: X\to \Bbb K^m$ with a finite number of zeroes.