The Leray–Schauder approach to the degree theory for -perturbations of maximal monotone operators in separable reflexive Banach spaces

The purpose of this paper is to demonstrate the fact that the topological degree theory of Leray and Schauder may be used for the development of the topological degree theory for bounded demicontinuous (S₊)-perturbations f of strongly quasibounded maximal monotone operators T in separable reflexive Banach spaces. Certain basic homotopy properties and the extension of this degree theory to (possibly unbounded) strongly quasibounded perturbations f are shown to hold. This work uses the well known embedding of Browder and Ton, and extends the work of Berkovits who developed this theory for the case T=0. Besides being an interesting mathematical problem, the existence of such a degree theory may, conceivably, become useful in situations where the use of the Leray–Schauder degree (via infinite dimensional compactness) is necessary.     

Pore-network study of the characteristic periods in the drying of porous materials

We study the periods that develop in the drying of capillary porous media, particularly the constant rate (CRP) and the falling rate (FRP) periods. Drying is simulated with a 3-D pore-network model that accounts for the effect of capillarity and buoyancy at the liquid-gas interface and for diffusion through the porous material and through a boundary layer over the external surface of the material. We focus on the stabilizing or destabilizing effects of gravity on the shape of the drying curve and the relative extent of the various drying periods. The extents of CRP and FRP are directly associated with various transition points of the percolation theory, such as the breakthrough point and the main liquid cluster disconnection point. Our study demonstrates that when an external diffusive layer is present, the constant rate period is longer.     

Pentanuclear complexes with unusual structural topologies from the initial use of two aliphatic amino-alcohol ligands in Fe chemistry

Five novel pentanuclear Fe(3+) clusters with the aliphatic amino-alcohol ligands 3-amino-1-propanol (Hap) and 2-(hydroxymethyl)piperidine (Hhmpip) [Fe(5)(μ(3)-Ο)(2)(L)(4)(O(2)CR)(7)] [L = ap(-), R = Ph (1); L = ap(-), R = C(CH(3))(3) (2); L = hmpip(-), R = Ph (3); L = hmpip(-), R = C(CH(3))(3) (4)] and [Fe(5)(μ(4)-Ο)(μ(3)-Ο)(O(2)CC(CH(3))(3))(8)(ap)(2)Cl(HO(2)CC(CH(3))(3))] (5) are reported. Compounds 1-4 were prepared from reactions of preformed trinuclear Fe(3+) clusters with the ligands in a molar ratio 1 : 5 in MeCN (1, 3, 4) or DMF (2), whereas compound 5 was prepared from the reaction of FeCl(3) with Hap in the presence of HO(2)CC(CH(3))(3) in a molar ratio 1 : 3 : 2 in MeCN. To the best of our knowledge, 1-5 are the first examples of Fe(3+) complexes with the ligands Hap and Hhmpip. The structures of 1-4 are composed of a quasi-planar [Fe(5)(μ(3)-O)(2)](11+) core which consists of two vertex-sharing [Fe(3)(μ(3)-O)](7+) triangles. The structure of 5 is based on the [Fe(5)(μ(4)-O)(μ(3)-O)](11+) core, in which the five Fe(3+) ions adopt a monocapped trigonal pyramidal topology. Variable-temperature magnetic susceptibility measurements on powdered microcrystalline samples of 1 and 5 revealed the existence of antiferromagnetic interactions which led to an S = 5/2 ground state. M?ssbauer spectroscopy studies on powdered microcrystalline samples of 1 and 5 confirmed that all iron ions of both complexes are in the Fe(3+) (S = 5/2) state. The variation of the ligand environment in the various iron sites was reflected in their different quadruple splitting parameters. At T < 50 K the M?ssbauer spectra indicated the onset of spin relaxation effects in the time scale of the technique (10(-7)-10(-8) s).     

Synthesis and characterization of γ-Fe2O3/carbon hybrids and their application in removal of hexavalent chromium ions from aqueous solutions

Magnetic Fe(2)O(3)/carbon hybrids were prepared in a two-step process. First, acetic acid vapor interacted with iron cations dispersed on the surface of a nanocasted ordered mesoporous carbon (CMK-3). In the second step, the primarily created iron acetate species underwent pyrolysis and transformed to magnetic iron oxide nanoparticles. X-ray diffraction, Fourier-transform infrared, and Raman spectroscopies were used for the chemical and structural characterization of the hybrids, while surface area measurements, thermal analysis, and transmission electron microscopy were employed to determine their physical, surface, and textural properties. These results revealed the preservation of the host carbon structure, which was homogenously and controllably loaded (up to 27 wt %) with nanosized (ca. 20 nm) iron oxides inside the mesoporous system. M?ssbauer spectroscopy and magnetic measurements at low temperatures confirmed the formation of γ-Fe(2)O(3) nanoparticles exhibiting superparamagnetic behavior. The kinetic studies showed a rapid removal of Cr(VI) ions from the aqueous solutions in the presence of these magnetic mesoporous hybrids and a considerably increased adsorption capacity per unit mass of sorbent in comparison to that of pristine CMK-3 carbon. The results also indicate highly pH-dependent sorption efficiency of the hybrids, whereas their kinetics was described by a pseudo-second-order kinetic model. Taking into account the simplicity of the synthetic procedure and possibility of magnetic separation of hybrids with immobilized pollutant, the developed mesoporous nanomaterials have quite real potential for applications in water treatment technologies.     

Stereoselective synthesis of α- C-allyl-glycopyranosides

A simplified procedure for the synthesis of C-glycosides has been developed. Fully or partially acetylated glycopyranoses are reacted, in a single step, with allyltrimethylsilane in the presence of a Lewis acid. This method also offers the advantage that stereoselectivity can be induced by using appropriate solvents.     

Nonresonance problems for differential inclusions in separable Banach spaces

Let be a real separable Banach space. The boundary value problem

is studied on the infinite interval Here, the closed and densely defined linear operator generates an evolution operator The function is measurable in its first variable, upper semicontinuous in its second and has weakly compact and convex values. Either is bounded and is compact for or is compact and is equicontinuous. The mapping is a bounded linear operator and is fixed. The nonresonance problem is solved by using Ma's fixed point theorem along with a recent result of Przeradzki which characterizes the compact sets in

     

New results in the perturbation theory of maximal monotone and -accretive operators in Banach spaces

Let be a real Banach space and a bounded, open and convex subset of The solvability of the fixed point problem in is considered, where is a possibly discontinuous -dissipative operator and is completely continuous. It is assumed that is uniformly convex, and A result of Browder, concerning single-valued operators that are either uniformly continuous or continuous with uniformly convex, is extended to the present case. Browder's method cannot be applied in this setting, even in the single-valued case, because there is no class of permissible homeomorphisms. Let The effect of a weak boundary condition of the type on the range of operators is studied for -accretive and maximal monotone operators Here, with sufficiently large norm and Various new eigenvalue results are given involving the solvability of with respect to Several results do not require the continuity of the operator Four open problems are also given, the solution of which would improve upon certain results of the paper.