$${\varvec{\mathcal {}}}$$-minimaxness and local cohomology modules

Let R be a commutative Noetherian ring, \({\mathfrak {a}}\) an ideal of R, M a finitely generated R-module, and \({\mathcal {S}}\) a Serre subcategory of the category of R-modules. We introduce the concept of \({\mathcal {S}}\)-minimax R-modules and the notion of the \({\mathcal {S}}\)-finiteness dimension

$$\begin{aligned} f_{\mathfrak {a}}^{{\mathcal {S}}}(M):=\inf \lbrace f_{\mathfrak {a}R_{\mathfrak {p}}}(M_{\mathfrak {p}}) \vert \mathfrak {p}\in {\text {Supp}}_R(M/ \mathfrak {a}M) \text { and } R/\mathfrak {p}\notin {\mathcal {S}} \rbrace \end{aligned}$$

and we will prove that: (i) If \({\text {H}}_{\mathfrak {a}}^{0}(M), \cdots ,{\text {H}}_{\mathfrak {a}}^{n-1}(M)\) are \({\mathcal {S}}\)-minimax, then the set \(\lbrace \mathfrak {p}\in {\text {Ass}}_R( {\text {H}}_{\mathfrak {a}}^{n}(M)) \vert R/\mathfrak {p}\notin {\mathcal {S}}\rbrace \) is finite. This generalizes the main results of Brodmann–Lashgari (Proc Am Math Soc 128(10):2851–2853, 2000), Quy (Proc Am Math Soc 138:1965–1968, 2010), Bahmanpour–Naghipour (Proc Math Soc 136:2359–2363, 2008), Asadollahi–Naghipour (Commun Algebra 43:953–958, 2015), and Mehrvarz et al. (Commun Algebra 43:4860–4872, 2015). (ii) If \({\mathcal {S}}\) satisfies the condition \(C_{\mathfrak {a}}\), then

$$\begin{aligned} f_{\mathfrak {a}}^{{\mathcal {S}}}(M)= \inf \lbrace i\in {\mathbb {N}}_{0} \vert {\text {H}}_{\mathfrak {a}}^{i}(M) \text { is not } {\mathcal {S}}\hbox {-}minimax\rbrace . \end{aligned}$$

This is a formulation of Faltings’ Local-global principle for the \({\mathcal {S}}\)-minimax local cohomology modules. (iii) \( \sup \lbrace i\in {\mathbb {N}}_{0} \vert {\text {H}}_{\mathfrak {a}}^{i}(M) \text { is not } {\mathcal {S}}\text {-minimax} \rbrace = \sup \lbrace i\in {\mathbb {N}}_{0} \vert {\text {H}}_{\mathfrak {a}}^{i}(M) \text { is not in } {\mathcal {S}} \rbrace \).

     

Minimaxness and finiteness properties of local homology and local cohomology modules

We prove some results concerning minimaxness and finiteness of local homology modules and by Matlis duality we extend some results for the minimaxness and finiteness of local cohomology modules. We introduce the concept of C-minimax R-modules, and we discuss the maximum and minimum integers such that local homology and local cohomology modules are C-minimax. As a consequence, we find minimum integers such that local homology and local cohomology modules are of finite length.     

On the asymptotic properties of non-autonomous systems

By giving a new method, we study asymptotic behavior of weakly almost nonexpansive sequences and curves introduced by Djafari Rouhani (J Differ Equ 229:412–425, 2006) in a reflexive Banach space X. Subsequently, we apply our results to study the asymptotic properties of unbounded trajectories for the quasi-autonomous dissipative system ${du/dt +Au\ni f}By giving a new method, we study asymptotic behavior of weakly almost nonexpansive sequences and curves introduced by Djafari Rouhani (J Differ Equ 229:412–425, 2006) in a reflexive Banach space X. Subsequently, we apply our results to study the asymptotic properties of unbounded trajectories for the quasi-autonomous dissipative system du/dt +Au ' f{du/dt +Au\ni f}, where A is an accretive (possibly multivalued) operator in X × X, and for some f_￥ ? X{f_{\infty}\in X} and 1 ≤ p < ∞ we have g ? L^p((1,+￥);X){g\in L^p((1,+\infty);X)}, so that g(θ) = (f(θ) − f _∞)/θ, ${(\forall \theta > 1)}${(\forall \theta > 1)}. Our results extend and improve many previously known results. Moreover, we answer an open question raised by B. Djafari Rouhani.     

Microwave-assisted rapid and efficient deprotection and direct esterification and silylation of MOM and EOM ethers catalyzed by [Hmim][HSO4] as a Br?nsted acidic ionic liquid

Abstract  

1-Methylimidazolium hydrogensulfate, [Hmim][HSO₄], a Br?nsted acidic room temperature ionic liquid, is used as a catalyst and reaction medium for facile and eco-friendly deprotection of methoxymethyl (MOM) and ethoxymethyl (EOM) ethers to their corresponding alcohols under thermal conditions (Δ) and microwave irradiation (MW). Furthermore, one-pot interconversion to the respective acetates and trimethylsilyl (TMS) ethers was also achieved.     

Application of surfactant assisted dispersive liquid-liquid microextraction for sample preparation of chlorophenols in water samples

A simple, rapid, and efficient method, based on surfactant assisted dispersive liquid-liquid microextraction (SA-DLLME), followed by high performance liquid chromatography (HPLC) has been developed for the extraction and determination of chlorophenols as model compounds in environmental water samples. A conventional cationic surfactant called cethyltrimethyl ammonium bromide (CTAB) was used as a disperser agent in the proposed approach. Thirty-five microliter of 1-octanol as an extraction solvent was injected rapidly into 11 mL aqueous sample containing 0.09 mmol L⁻¹ of CTAB, the mixture was then shaken for 3 min to disperse the organic phase. Having the extraction procedure been completed, the mixture was centrifuged and 20 μL of collected phase was injected into HPLC for subsequent analysis. Some parameters such as the type and volume of the extraction solvent, the type and concentration of surfactant, pH, ionic strength, shaking time, extraction temperature and centrifugation time were optimized. The preconcentration factors (PFs) in a range of 187-353 were obtained under the optimum conditions. The linear range, detection limit (S/N = 3), and precision (n = 5) were 0.2-200, 0.1 μg L⁻¹, and 4.7-6.9%, respectively. Tap water, sea water and mineral water samples were successfully analyzed for the existence of chlorophenols using the proposed method.     

H3PW12O40–[bmim][FeCl4]: A novel and green catalyst-medium system for microwave-promoted selective interconversion of alkoxymethyl ethers into their corresponding nitriles, bromides and iodides

In the present work, the catalytic activity of 12-tungstophosphoric acid immobilized on [bmim][FeCl₄] ionic liquid as a highly efficient and eco-friendly catalytic system for rapid and chemoselective direct conversion of MOM- or EOM-ethers into their corresponding nitriles, bromides and iodides under microwave irradiation is reported. In these reactions, the products are obtained in high yields. The catalyst exhibited remarkable reactivity and was reused several times.     

Molybdenum hexacarbonyl supported on amine modified multi‐wall carbon nanotubes: an efficient and highly reusable catalyst for epoxidation of alkenes with tert‐butylhydroperoxide

In the present work, highly efficient epoxidation of alkenes catalyzed by Mo(CO)₆ supported on multi‐wall carbon nanotubes modified by 2‐aminopyrazine, APyz‐MWCNTs, is reported. The prepared catalyst was characterized by elemental analysis, scanning electron microscopy, FT IR and diffuses reflectance UV–vis spectroscopic methods. This new heterogenized catalysts, [Mo(CO)₆@APyz‐MWCNT], was used as a highly efficient catalyst for epoxidation of alkenes with tert‐BuOOH. This robust catalyst was reused several times without loss of its catalytic activity. Copyright © 2010 John Wiley & Sons, Ltd.     

The Synthesis of Ahyl Diazomethanes

A conyenient and rapid method for preparing aryldiazo-methanes from the related tosylhydrazones employing phase transfer catalysis with hydrocarbon solvents in good to excellent yields has been developed.     

Novel thermally stable poly(sulfone ether ester imide)s

A novel sulfone ether ester diamine was prepared by a three-step method. Reaction of 1,5-dihydroxy naphthalene with 4-nitrobenzoyl chloride afforded 5-hydroxy-1-naphthyl-4-nitrobenzoate. Reduction of nitro group by iron powder and HCl resulted in preparation of 5-hydroxyl-1-naphthyl-4-aminobenzoate. Reaction of this compound (two moles) with bis (4-chlorophenyl) sulfone led to preparation of a novel sulfone ether ester diamine. Three novel aromatic poly(sulfone ether ester imide)s were synthesized by polycondensation reactions of the prepared diamine with aromatic dianhydrides. Conventional methods were used to characterize the structure of the monomer and polymers. Physical properties of the polymers were also studied. The polyimides showed high thermal stability.     

Novel thermally stable polyimides based on flexible diamine: synthesis, characterization, and properties

A novel diamine with built-in sulfone, ether, and amide structure was prepared via three-step reactions. Nucleophilic reaction of 4-aminophenol with 4-nitrobenzoyl chloride in the presence of propylene oxide led to preparation of N-(4-hydroxy phenyl)-4-nitrobenzamide (HPNB). The nitro group of this compound was reduced with hydrazine and Pd/C to afford 4-amino-N-(4-hydroxy phenyl)benzamide (AHPB). Two moles of AHPB were reacted with bis-(4-chloro phenyl)sulfone to provide a novel sulfone ether amide diamine (SEAD). All the prepared compounds were characterized by common spectroscopic methods. The prepared diamine (SEAD) used to prepare related polyimides by reaction with different aromatic dianhydrides. The obtained poly(sulfone ether amide imide)s were characterized and their properties were studied.