An Oxidative Desulfurization Catalyst Based on Bimodal Mesoporous Silica Containing Quaternary Ammonium Heteropolyphosphamolybdenum

A bimodal mesoporous silica(BMMS) modified with amphiphilic compound(C_(19)H_(42)N)_3(PMo_(12)O_(40))(CTA-PMO) was prepared by the two-step impregnation method. Firstly, H3PMo12O40 was introduced into the bimodal mesoporous silica via impregnation, then C_(19)H_(42)NBr(CTAB) was grafted on the surface of BMMS containing H3PMo12O40 based on the chemical reaction between quaternary ammonium compound and the phosphomolybdic acid, and then the catalyst CTAPMO/BMMS was obtained. The samples were characterized by XRD, N_2 adsorption and desorption, FTIR, 31P-NMR, 29Si-NMR and TEM analyses. It is shown that the catalyst has a typical bimodal mesoporous structure, in which the small mesopore diameter is about 3.0 nm and the large mesopore diameter is about 5.0 nm. The chemical interaction happens between the Keggin structure and silica group of BMMS. Compared with the mono-modal porous Hβ and SBA-15 zeolites modified with CTA-PMO, CTA-PMO/BMMS showed better catalytic activity in the oxidative conversion of dibenzothiophene(DBT), and the desulfurization rate can reach about 94% with the help of extraction, and the catalyst can be separated by filtration and reused directly. The catalytic oxidative desulfurization mechanism on CTA-PMO/BMMS was proposed and verified.

Biorthogonal multiple wavelets generated by vector refinement equation

Biorthogonal multiple wavelets are generated from refinable function vectors by using the multiresolution analysis. In this paper we provide a general method for the construction of compactly supported biorthogonal multiple wavelets by refinable function vectors which are the solutions of vector refinement equations of the form $$\varphi (x) = \sum\limits_{\alpha \in \mathbb{Z}^s } {a(\alpha )\varphi (Mx - \alpha ), x \in \mathbb{R}^s } ,$$ where the vector of functions ? = (? 1, …, ? r)T is in $(L_2 (\mathbb{R}^s ))^r ,a = :(a(\alpha ))_{\alpha \in \mathbb{Z}^s } $ is a finitely supported sequence of r × r matrices called the refinement mask, and M is an s × s integer matrix such that lim n→∞ M ?n = 0. Our characterizations are in the general setting and the main results of this paper are the real extensions of some known results.

Gabor systems on discrete periodic sets

Due to its good potential for digital signal processing, discrete Gabor analysis has interested some mathematicians. This paper addresses Gabor systems on discrete periodic sets, which can model signals to appear periodically but intermittently. Complete Gabor systems and Gabor frames on discrete periodic sets are characterized; a sufficient and necessary condition on what periodic sets admit complete Gabor systems is obtained; this condition is also proved to be sufficient and necessary for the existence of sets E such that the Gabor systems generated by χE are tight frames on these periodic sets; our proof is constructive, and all tight frames of the above form with a special frame bound can be obtained by our method; periodic sets admitting Gabor Riesz bases are characterized; some examples are also provided to illustrate the general theory.

Riesz multiwavelet bases generated by vector refinement equation

In this paper, we investigate compactly supported Riesz multiwavelet sequences and Riesz multiwavelet bases for L 2(? s ). Suppose ψ = (ψ1,..., ψ r ) T and $ \tilde \psi = (\tilde \psi ^1 ,...,\tilde \psi ^r )^T $ are two compactly supported vectors of functions in the Sobolev space (H μ(? s )) r for some μ > 0. We provide a characterization for the sequences {ψ jk l : l = 1,...,r, j ε ?, k ε ? s } and $ \tilde \psi _{jk}^\ell :\ell = 1,...,r,j \in \mathbb{Z},k \in \mathbb{Z}^s $ to form two Riesz sequences for L 2(? s ), where ψ jk l = m j/2ψ l (M j ·?k) and $ \tilde \psi _{jk}^\ell = m^{{j \mathord{\left/ {\vphantom {j 2}} \right. \kern-0em} 2}} \tilde \psi ^\ell (M^j \cdot - k) $ , M is an s × s integer matrix such that lim n→∞ M ?n = 0 and m = |detM|. Furthermore, let ? = (?1,...,? r ) T and $ \tilde \phi = (\tilde \phi ^1 ,...,\tilde \phi ^r )^T $ be a pair of compactly supported biorthogonal refinable vectors of functions associated with the refinement masks a, $ \tilde a $ and M, where a and $ \tilde a $ are finitely supported sequences of r × r matrices. We obtain a general principle for characterizing vectors of functions ψν = (ψν1,...,ψνr ) T and $ \tilde \psi ^\nu = (\tilde \psi ^{\nu 1} ,...,\tilde \psi ^{\nu r} )^T $ , ν = 1,..., m ? 1 such that two sequences {ψ jk νl : ν = 1,..., m ? 1, l = 1,...,r, j ε ?, k ε ? s } and { $ \tilde \psi _{jk}^\nu $ : ν=1,...,m?1,?=1,...,r, j ∈ ?, k ∈ ? s } form two Riesz multiwavelet bases for L 2(? s ). The bracket product [f, g] of two vectors of functions f, g in (L 2(? s )) r is an indispensable tool for our characterization.

Necessary conditions for Gabor frames

The goal of this paper is to establish a set of necessary conditions for Gabor frames. These conditions are also sufficient for tight frames.