The existence of slowly and regularly varying solutions in the sense of Karamata implying nonoscillation is proved for a class of second order nonlinear retarded functional differential equations of Thomas-Fermi type. A motivation for such study is the extensively developed theory offering a number of properties of regularly and slowly varying functions ([2]) - consequently of such solutions of differential equations. As an illustration, the precise asymptotic behaviour for $t\rightarrow \infty$ of the slowly varying solutions for a subclass of considered equations is presented.

We show that the Lebesgue spaces for defining BMO can be replaced by $p$-convex rearrangement-invariant quasi-Banach function spaces associated with $A_{p}$-weighted measures.

We consider the Rational Large Eddy Simulation (RLES) model introduced by Galdi and Layton (Math. Models Methods Appl. Sci. 10 (2000) 343-350). Various regularity criteria for the strong solution of this model are established here, which improve previous ones.

We are interested in a system of Stokes type, where the divergence-free constraint is modified by adding a term proportional to the pressure. The domain is the half-space with nonhomogeneous Navier's boundary conditions. The weighted Sobolev spaces yield a natural functional framework to envisage a wide class of behavior at infinity for data and solutions. So, we can give a range of solutions from strong to very weak depending on the regularity of the data. All along this study, we take the bridge between this system and the linear elasticity system.

Ravenel introduced spectra $T(m)$ for $m\ge 0$ interpolating the Brown-Peterson spectrum $BP$ and the sphere spectrum $S$ in Complex cobordism and stable homotopy groups of spheres, AMS Chelsea Publishing, Providence, 2004. Since the homotopy groups of $BP$ are well known, it is interesting to study differences among the homotopy groups of $T(m)$'s to study the homotopy groups of spheres. He also introduced the localization functor $L_n$ on the stable homotopy category in "Localization with respect to certain periodic homology theories," Amer. J. Math. 106 (1984), 351–414. To study the difference of $L_nT(m)$'s for a fixed integer $n$, we consider the corresponding chromatic $E_1$-term $\e^0(M^1_{n-1})$ on $\Ga(m+1)$ for each $m$, and determine it for $m+1\ge (n-2)(n-1)$ in this paper. The results show that the structures depend on a integer $\Lt[\dfrac{m+1}{n-1}\Rt]$. Here $[x]$ denotes the greatest integer that does not exceed $x$.

An efficient application of capacitary functions for compact subsets of the Royden harmonic boundary to an inverse inclusion problem concerning spaces of Dirichlet finite and mean bounded harmonic functions in the classification theory of Riemann surfaces is given.

On Sendov's conjecture, V. Vâjâitu and A. Zaharescu (and M. J. Miller, independently) state the following in their paper: if one zero $a$ of a polynomial which has all the zeros in the closed unit disk is sufficiently close to the unit circle, then the distance from $a$ to the closest critical point is less than $1$. It is desirable to quantify this assertion. In the author's previous paper, we obtained an upper bound on the radius of the disk centered at the origin which contains all the critical points. In this paper, we improve it, and then, estimate the range of the zero $a$ satisfying the above. This result, moreover, implies that if a zero of a polynomial is close to the unit circle and all the critical points are far from the zero, then the polynomial must be close to $P(z)=z^n-c$ with $\abs{c}=1$.