Volume 101, Issue 2
August 1991, pages 63-147
pp 63-70 August 1991
On the convergence properties of measurable multifunctions with values in a separable Banach space
In this paper we prove some convergence theorems for Banach space valued multifunctions. First we consider the notion of weak convergence of sets and prove a weak completeness and a weak compactness result of the Dunford-Pettis type for weakly compact, convex valued integrable multifunctions. Then we consider set valued martingales and establish two convergence theorems. One using the Kuratowski-Mosco mode of convergence and for the other the Hausdorff mode.
pp 71-86 August 1991
Uniformly non-l_{n}^{(1)} Musielak-Orlicz sequence spaces
We give a necessary and sufficient condition for the uniformly non-l_{n}^{(1)} property of Musielak-Orlicz sequence spacesl^{Φ} generated by a sequence Φ=(ϕ_{n}:n⩾l) of finite Orlicz functions such that$$\mathop {\lim }\limits_{u \to 0} u^{ - 1} \phi _n (u) = 0$$ for eachn∈ℕ. As a result, forn_{0}⩾2, there exist spacesl^{Φ} which are only uniformly non-l_{n}^{(1)}forn⩾n_{0}. Moreover we obtain a characterization of uniformly non-l_{n}^{(1)} and reflexive Orlicz sequence spaces over a wide class of purely atomic measures and of uniformly non-l_{n}^{(1)} Nakano sequence spaces. This extends a result of Luxemburg in [19].
pp 87-110 August 1991
Congruence subgroup problem for anisotropic groups over semilocal rings
In Chapter I, a theorem of Margulis which gives the structure of normal subgroups ofSL(1,D) for a quaternion division algebraD over a global fieldK of characteristic not 2, is generalized to semi-local ringsR inK. Using this, we obtain in Chapter II, a description of normal subgroups ofG(R) forK-anisotropic algebraic groupsG of typesA_{3}, B_{n}, C_{n},^{1}D_{n},^{2}D_{n} and some forms of^{2}A_{n}. As a Corollary, a proof of the Platonov-Margulis conjecture is obtained for the above groups.
pp 111-120 August 1991
On continuous maps between Grassmann manifolds
LetG_{n,k} denote the Grassmann manifold ofk-planes in ℝ^{n}. We show that for any continuous mapf: G_{n,k}→G_{n,l} the induced map inZ/2-cohomology is either zero in positive dimensions or has image in the subring generated by w_{1}(γ_{n, k}), provided 1≤l<k≤[n/2] andn≥k+2l-1. Our main application is to obtain negative results on the existence of equivariant maps between oriented Grassmann manifolds. We also obtain positive results in many cases on the existence of equivariant maps between oriented Grassmann manifolds.
pp 121-125 August 1991
On integrability of power series
This paper deals with the integrability of a power series. Our results generalize certain results of Ram, and Askey and Karlin.
pp 127-141 August 1991
Linear, hydrodynamic flow in a rotating saturated porous medium
Linear, steady, axisymmetric flow of a homogeneous fluid in a rigid, bounded, rotating, saturated porous medium is analyzed. The fluid motions are driven by differential rotation of horizontal boundaries. The dynamics of the interior region and vertical boundary layers are investigated as functions of the Ekman number E(=v/ΩL^{2}) and rotational Darcy 3 numberN(=kΩ/v) which measures the ratio between the Coriolis force and the Darcy frictional term. IfN≫E^{−1/2}, the permeability is sufficiently high and the flow dynamics are the same as those of the conventional free flow problem with Stewartson'sE^{1/3} andE^{1/4} double layer structure. For values ofN≤E^{−1/2} the effect of porous medium is felt by the flow; the Taylor-Proudman constraint is no longer valid. ForN≪E^{−1/3} the porous medium strongly affects the flow; viscous side wall layer is absent to the lowest order and the fluid pumped by the Ekman layer, returns through a region of thicknessO(N^{−1}). The intermediate rangeE^{−1/3}≪N≪E^{−1/2} is characterized by double side wall layer structure: (1)E^{1/3} layer to return the mass flux and (ii) (NE)^{1/2} layer to adjust the interior azimuthal velocity to that of the side wall. Spin-up problem is also discussed and it is shown that the steady state is reached quickly in a time scaleO(N).
pp 143-146 August 1991
In this paper a theorem on$$\left| {\bar N,p_n } \right|_k $$ summability factors of infinite series, which generalizes a theorem of Bor [2], has been proved.
pp 147-147 August 1991 Erratum
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