We examine the bifurcation curves of solutions to the Kolmogorov problem and present the exact formula for the second derivatives of their components concerning Reynolds numbers at bifurcation points. Using this formula, we show the supercriticality of these curves in the case where the ratio of periodicities in two directions is close to one. In order to prove this, we construct an inverse matrix of infinite order, whose elements are given by sequences generated by continued fractions. For this purpose, we investigate some fundamental properties of these sequences such as quasi-monotonicity and exponential decay from general viewpoints.

Germs of holomorphic vector fields at the origin $0\in \co^{\kern1pt2}$ and polynomial vector fields on $\co^{\kern1pt2}$ are studied. Our aim is to classify these vector fields whose orbits have bounded geometry in a certain sense. Namely, we consider the following situations: (i) the volume of orbits is an integrable function, (ii) the orbits have sub-exponential growth, (iii) the total curvature of orbits is finite. In each case we classify these vector fields under some generic hypothesis on singularities. Applications to questions, concerning polynomial vector fields having closed orbits and complete polynomial vector fields, are given. We also give some applications to the classical theory of compact foliations.

We establish various new boundedness results for a class of nonlinear planar systems including some generalized Liénard equations. These results represent significant improvement and generalization of many existing ones in the literature. Our sufficient conditions are sharp in the sense that for some special but quite general cases, they coincide with the necessary conditions. Three illustrative examples are given.

Floquet multipliers of symmetric rapidly oscillating periodic solutions of the differential delay equation $\dot x(t)=\alpha f(x(t),x(t-1))$ with the symmetries $ f(-x,y)=f(x,y)=-f(x,-y)$ are described in terms of zeroes of a characteristic function. A relation to the characteristic function of symmetric slowly oscillating periodic solutions is found. Sufficient conditions for the existence of at least one real multiplier outside the unit disc are derived. An example with a piecewise linear function $f$ is studied in detail, both analytically and numerically.

For a Markov process associated with a not necessarily symmetric regular Dirichlet form, if the form satisfies the sector condition, then any semipolar sets are exceptional. On the other hand, in the case of the space-time Markov process associated with a family of time dependent Dirichlet forms, there exist non-exceptional semipolar sets. The main purpose of this paper is to show that any semipolar set $B=J\times \Gamma$ of the direct product type of a subset $J$ of time and a subset $\Gamma$ of space is exceptional if $J$ has positive Lebesgue measure.

We use Cox's description for sheaves on toric varieties and results about local cohomology with respect to monomial ideals to give a characteristic-free approach to vanishing results on toric varieties. As an application, we give a proof of a strong version of Fujita's Conjecture in the case of toric varieties. We also prove that every sheaf on a toric variety corresponds to a module over the homogeneous coordinate ring, generalizing Cox's result for the simplicial case.