Mixed‐Metal MIL‐100(Sc,M) (M=Al,Cr, Fe) for Lewis Acid Catalysis and Tandem CC Bond Formation and Alcohol Oxidation

The trivalent metal cations Al³⁺, Cr³⁺, and Fe³⁺ were each introduced, together with Sc³⁺, into MIL‐100(Sc,M) solid solutions (M=Al, Cr, Fe) by direct synthesis. The substitution has been confirmed by powder X‐ray diffraction (PXRD) and solid‐state NMR, UV/Vis, and X‐ray absorption (XAS) spectroscopy. Mixed Sc/Fe MIL‐100 samples were prepared in which part of the Fe is present as α‐Fe₂O₃ nanoparticles within the mesoporous cages of the MOF, as shown by XAS, TGA, and PXRD. The catalytic activity of the mixed‐metal catalysts in Lewis acid catalysed Friedel–Crafts additions increases with the amount of Sc present, with the attenuating effect of the second metal decreasing in the order Al>Fe>Cr. Mixed‐metal Sc,Fe materials give acceptable activity: 40 % Fe incorporation only results in a 20 % decrease in activity over the same reaction time and pure product can still be obtained and filtered off after extended reaction times. Supported α‐Fe₂O₃ nanoparticles were also active Lewis acid species, although less active than Sc³⁺ in trimer sites. The incorporation of Fe³⁺ into MIL‐100(Sc) imparts activity for oxidation catalysis and tandem catalytic processes (Lewis acid+oxidation) that make use of both catalytically active framework Sc³⁺ and Fe³⁺. A procedure for using these mixed‐metal heterogeneous catalysts has been developed for making ketones from (hetero)aromatics and a hemiacetal.     

Zeolites with Continuously Tuneable Porosity

Zeolites are important materials whose utility in industry depends on the nature of their porous structure. Control over microporosity is therefore a vitally important target. Unfortunately, traditional methods for controlling porosity, in particular the use of organic structure‐directing agents, are relatively coarse and provide almost no opportunity to tune the porosity as required. Here we show how zeolites with a continuously tuneable surface area and micropore volume over a wide range can be prepared. This means that a particular surface area or micropore volume can be precisely tuned. The range of porosity we can target covers the whole range of useful zeolite porosity: from small pores consisting of 8‐rings all the way to extra‐large pores consisting of 14‐rings.     

Uniqueness of highly representative surface embeddings

Let Σ be a (connected) surface of “complexity” κ; that is, Σ may be obtained from a sphere by adding either ½κ handles or κ crosscaps. Let ρ ≥ 0 be an integer, and let Γ be a “ρ-representative drawing” in Σ; that is, a drawing of a graph in Σ so that every simple closed curve in Σ that meets the drawing in < ρ points bounds a disc in Σ. Now let Γ′ be another drawing, in another surface Σ′ of complexity κ′, so that Γ and Γ′ are isomorphic as abstract graphs. We prove that. (i) If ρ ≥ 100 log κ/ log log κ (or ρ ≥ 100 if κ ≤ 2) then κ′ ≥ κ, and if κ′ = κ and Γ is simple and 3-connected there is a homeomorphism from Σ to Σ′ taking Γ to Γ′, and. (ii) if Γ is simple and 3-connected and Γ′ is 3-representative, and ρ ≥ min (320, 5 log κ), then either there is a homeomorphism from Σ to Σ′ taking Γ to Γ′, or κ′ ≥ κ + 10^-4 ρ². © 1996 John Wiley & Sons, Inc.     

Nonlinear two-point boundary value problems with multiple solutions

In the first part of this paper we study the convergence of finite difference methods to approximate the maximal solution of problems of the form:u+f (x, u)=0, with boundary conditions eitheru(0)=u(b)=0 oru(0)=u (b)=0, 0<b1. The functionf(x, u) satisfies several conditions that are explicitly given in § 1. This work extends earlier results of Parter (see references at the end).Since this problem has in general more than one solution we develop in the second part two algorithms to approximate solutions characterized by the number of their zeros in (0, 1). We include in the last Section numerical results and some additional comments on the implementation of the algorithms on a digital computer.The research of this author was supported by the Office of Naval Research under contract N^o N-0014-67-A-0128-004.Part of this research was carried on while this author was visiting the Mathematics Research Center, University of Wisconsin, during the summer of 1969.     

The calculation of resonance separatrices for the near-integrable case of the standard mapping

The standard mapping arises in many physical applications, including the analysis of nonlinear resonant acoustic oscillations in a closed tube. A perturbation expansion, in powers of the amplitude parameter, is given for the calculation of the fixed points of various orders and the associated separatrices. It is shown how exact homoclinic orbits can be calculated numerically. Explicit analytic expressions are given for the separatrices associated with the first four resonances when the perturbation parameter is small.     

Fourier analysis on the Heisenberg group. I. Schwartz space

We derive a usable characterization of the group FT (Fourier Transform) of Schwartz space on the Heisenberg group $\text{H}$ⁿ, in terms of certain asymptotic series. To accomplish this we study in detail the FT of multiplication and differentiation operators on $\text{H}$ⁿ, the relation of multiple Fourier series to the FT, and the process of group contraction on $\text{H}$ⁿ. We use our characterization to solve a form of the division problem for convolution of $\text{H}$ⁿ, which has application to Hardy space theory.