Method of separation of the variables for basic analogs of equations of mathematical physics

We apply method of separation of the variables to q-analogs of several equations of mathematical physics. Dedicated to Dick Askey on his 70th birthday. 2000 Mathematics Subject Classification Primary—33D45, 42C10; Secondary—33D15.     

Decolorization of vegetable oils: chlorophyll-a adsorption by acid-activated sepiolite

The adsorption characteristics of acid activated sepiolite (AAS) for the removal of chlorophyll-a (C(55)H(72)MgN(4)O(5)) from rapeseed oil was studied as a function of different sepiolite dosages and bleaching temperatures. A correlation has been shown between the adsorption capacity and a combination of AAS amount, bleaching temperature and oxidative reactions of chlorophyll-a. The adsorption equilibrium was found to follow the Langmuir isotherm model with a maximum adsorption capacity of 0.36 mg/g on AAS and K(L) value ranging from 6.11 to 19.51 kg/mg at 80 and 100 degrees C. It was found that AAS is an effective sorbent for the removal of chlorophyll-a molecule which is believed to adsorb as a protonated species onto SiOH groups at the edge in the tetrahedral sheet of sepiolite. These findings reveal that chlorophyll-a molecules not only adsorb onto the external surface of AAS but replace the released Mg(2+) ions in the octahedral sheet; they, depending on the pore size of AAS, are also incorporated in the channels and tunnels of sepiolite. A structural model is proposed to account for the orientation of chlorophyll-a in the sepiolite matrix.     

Thermal degradation of two different polymers bearing amide pendant groups prepared by ATRP method

Piperidinocarbonylmethyl methacrylate (PyCMMA) and 1-(piperidinocarbonyl) ethylmethacrylate (PyCEMA) monomers were synthesized. Polymerizations of PyCMMA and PyCEMA were carried out by atom transfer radical polymerization. The structure of monomers and polymers was characterized by ¹H-NMR, ¹³C-NMR, and FT-IR spectroscopies. Characterization of poly(PyCMMA) and poly(PyCEMA) were carried out using differential scanning calorimetry and gel permeation chromatography. The experimental results showed that the reaction exhibited characteristics of controlled polymerization. The thermal degradation behaviors of poly(PyCEMA) and poly(PyCMMA) were studied using thermogravimetry and a single line vacuum system consisting of a degradation tube with a condenser for product collection. The poly(PyCEMA) and poly(PyCMMA) were heated from ambient temperature to 325 and 500 °C, respectively. The products of degradation were collected as a cold ring fraction (CRF). The CRFs of degradation were investigated by means of IR, ¹HNMR, and GC-MS. For the degradation of both polymers, the major products of CRFs are piperidinocarbonyl methanol and 1,2-dipiperidino,1-oxo ethane. The GC-MS, IR, and NMR data showed that depolymerization below 325 °C to the corresponding monomer was not prominantin the thermal degradation of poly(PyCMMA). The mode of thermal degradation including formation of the major products was identified.     

Functionalization with silyl enol ethers,VI Zinc chloride mediated alkoxyalkylation of O-methyl-O-trimethylsilyl) keten acetals with 2-alkoxy-1,3-dioxolanes

O-Methyl-O-(trimethylsilyl) keten acetals2 were regioselectively alkoxyalkylated by 2-alkoxy-1,3-dioxolanes1 in the presence of zinc chloride. This method represents a good way for synthesis of -protected -keto carbonic esters4.a)Akgün E,Pindur U (1983) Chem-Ztg 107: 236; b)Akgün E,Pindur U (1983) 107: 237; c)Akgün E,Pindur U (1984) Synthesis 1984: 227; d)Akgün E,Pindur U (1984) Monatsh Chem 115: 587; e)Akgün E,Pindur U (1985) Liebigs Ann Chem 1985: 2472     

Asymptotic behavior of recursions via fixed point theory

In this paper we provide a formulation of initial value problems for (explicit and implicit) difference equations in terms of abstract equations in sequence spaces. They will be solved using appropriate fixed point theorems and we obtain quantitative attractivity properties.     

On a subsemigroup of the universal covering of Lie semigroups

Let \(G\) be a connected Lie group and \(S\) a generating Lie semigroup. An important fact is that generating Lie semigroups admit simply connected covering semigroups. Denote by \(\widetilde{S}\) the simply connected universal covering semigroup of \(S\) . In connection with the problem of identifying the semigroup \(\Gamma (S)\) of monotonic homotopy with a certain subsemigroup of the simply connected covering semigroup \(\widetilde{S}\) we consider in this paper the following subsemigroup $$\begin{aligned} \widetilde{S}_{L}=\overline{\left\langle \mathrm {Exp}(\mathbb {L} (S))\right\rangle } \subset \widetilde{S}, \end{aligned}$$ where \(\mathrm {Exp}:\mathbb {L}(S)\rightarrow S\) is the lifting to \( \widetilde{S}\) of the exponential mapping \(\exp :\mathbb {L}(S)\rightarrow S\) . We prove that \(\widetilde{S}_{L}\) is also simply connected under the assumption that the Lie semigroup \(S\) is right reversible. We further comment how this result should be related to the identification problem mentioned above.