A note on double lacunary statistical $$\sigma$$-convergence of fuzzy numbers

Quite recently, Sava? (Appl Math Lett 21:134–141, 2008), defined the lacunary statistical analogue for double sequence \(X=\{X_{k,l}\}\) of fuzzy numbers as follows: a double sequence \(X=\{X_{k,l}\}\) is said to be lacunary P-statistically convergent to \(X_{0}\) provided that for each \(\epsilon >0\)

$ P-\lim_{r,s}\frac{1}{h_{r,s}}\left | \{(k,l)\in I_{r,s}: d(X_{k,l },X_0)\geq \epsilon\}\right|= 0. $

In this paper we introduce and study double lacunary \(\sigma\)-statistical convergence for sequence of fuzzy numbers and also we get some inclusion theorems.

     

Activities of MFI-Supported Rhenium Catalysts for the Aromatization of Methane: Effect of Cationic Form of the Inorganic Carrier

MFI type inorganic carrier was used in two different cationic forms, hydrogen and calcium respectively. MFI-supported molybdenum and rhenium catalysts were prepared. The activities of the catalysts were compared for the aromatization reaction of methane. Higher activity values were attained with the catalysts supported on HZSM-5. Aromatics were also observed with the catalysts supported on CaZSM-5, despite their deficiency in acid sites. Highly dispersed rhenium is expected to be formed with the use of the inorganic carrier in calcium form. On the other hand, lower reaction rates were observed with rhenium supported on CaZSM-5, in spite of the improved dispersion of the active rhenium species on this catalyst. This was interpreted in terms of the critical role of the acid sites in the conversion of methane to aromatics, compared to the improved dispersion of the active metal.     

Investigating the influence of tool shape during FSW of aluminum alloy via CFD analysis

This paper describes the application of the CFD code, Comsol Multiphysics, to modeling the 3-D metal flow in friction stir welding of AA 2024-T3 aluminum alloy in order to investigate the influence of tool shape over the metal flow. Heat transfer and non-Newtonian flow equations were solved simultaneously. The results from the benchmark experiments found in the literature were used for validation purposes. Scrolled shoulders and threaded pins were given as kinematic boundary conditions. This made the computational problem an easy one. A welding engineer can predict the metal flow around the tool with different scrolls and threads under any welding conditions without making expensive experiments. Investigation of the velocity field before actual welding can save a lot of engineering hours.     

Effects of pretreatment and various operating parameters on permeate flux and quality during ultrafiltration of apple juice

The central composite design with a quadratic model was used to investigate the effect of temperature (20–40 °C) and transmembrane pressure (TMP) (100–300 kPa) as well as pretreatment with gelatin and bentonite (0:0–300:1500 mg L^?1) on permeate flux and fouling resistance during ultrafiltration of apple juice. The changes in some physicochemical properties were also investigated. Pretreatment of apple juice with gelatin and bentonite and an increase in the temperature and TMP significantly improved the permeate flux. In general, physicochemical properties of apple juice other than polyphenolics were not affected by the applied parameters. Using 150 mg L^?1 of gelatin and 750 mg L^?1 of bentonite for fining significantly reduced the total phenolic content and so did the fouling resistance. Fouling resistance also decreased with increasing temperature, and increasing pressure increased the fouling resistance.     

Synthesis and characterization of a titanium phosphate-tellurite glass for Faraday rotators

This work is focused on investigation of thermal, structural, optical, magnetic, and magneto-optical properties of novel titanium phosphate-tellurite glass applied as Faraday rotators. The glass belonging to the system 35Li₂O–10Al₂O₃–5TiO₂–45P₂O₅–5TeO₂ was prepared by a nonconventional wet route of raw materials processing, followed by melting-quenching-annealing steps. Some important physical properties of the investigated glass have been measured and calculated, providing knowledge related to glass compactness, electronic structure, glass forming capability, etc. XRD analysis evidenced an amorphous network structure of the investigated glass. The optical absorption in the Vis domain is mainly due to Ti³⁺ ions and Te₂ clusters formed during the glass melting process. A relatively low optical absorption is noticed over 600 nm that activates a significant Faraday magneto-optical effect. Photoluminescence bands in the blue, red, and infrared domains are observed, caused by Te₂ clusters formed during the glass melting process. The magnetization in dependency on applied magnetic field reveals a complex behavior of the glass, depending on temperature. Thus, it is found a ferromagnetic behavior up to 2000 Oe, a paramagnetic component up to 40 000 Oe, followed by a diamagnetic one over 40 000 Oe. Faraday rotation angle and Verdet constant values in the visible domain are correlated with the reduced TeO₂ content of the glass.     

Influence of T5 heat treatment on the microstructure and lubricated wear behavior of ternary ZnAl40Cu2 and quaternary ZnAl40Cu2Si2.5 alloys

A ternary ZnAl40Cu2 and a quaternary ZnAl40Cu2Si2.5 alloys were produced by permanent mold casting and subjected to T5 heat treatment at a temperature of 150 °C for 24 hours. The structural, mechanical and lubricated wear properties of these alloys were investigated in the as-cast and heat-treated conditions and the results were compared with those of SAE 65 (CuSn12) plain bearing bronze. Microstructure of the ternary alloy consisted of aluminum-rich α, eutectoid conversion product of α+η and ϵ phase located in the interdendritic channels. In addition to these phases, silicon particles were observed in the microstructure of the quaternary alloy. T5 heat treatment caused a considerable amount of reduction in the hardness, tensile strength and wear resistance of ZnAl40-based ternary and quaternary alloys, but improved their ductility and stability. These alloys in the as-cast and heat-treated conditions exhibited lower wear volume or higher wear resistance than SAE 65 bearing bronze. Among the experimental alloys, the optimum mechanical properties and wear performance were obtained from ZnAl40Cu2Si2.5 alloy in both as-cast and heat-treated conditions. Adhesion appeared to be the main wear mechanism for the ZnAl40-based alloys, but abrasion dominated the wear of SAE 65 bronze.     

Use of FIB to Study ZDDP Tribofilms

Focussed ion beam milling (FIB) followed by TEM has been used to study ZDDP tribofilms on rubbed steel surfaces. It has been found that the impact of high energy platinum and gallium ions during FIB causes significant morphological and structural changes to the uppermost 30–50 nm of a ZDDP tribofilm. This can be prevented by the low energy deposition of a quite thick gold layer prior to installation of the sample in the FIB facility. This problem, and its solution, have been quite widely reported in the non-tribology literature but have not previously been highlighted in the application of FIB to study tribological surfaces. It has also been found, using this gold pre-deposition method, that the bulk of the ZDDP tribofilm studied has a polycrystalline structure.     

Study of thermal behavior of polyimides containing pendent-substituted azobenzene units

Thermal stability and degradation kinetic of the polyimides containing pendent-substituted azobenzene units were investigated by thermogravimetric analysis in nitrogen at different heating rates. Freeman–Carroll, Coats–Redfern and Flynn–Wall–Ozawa methods were utilized to obtain the kinetic parameters (activation energy E _a, pre-exponential factor A and order of reaction n) or variation of activation energy with conversion degree from thermogravimetric and derivative thermogravimetric curves. The thermogravimetric analysis revealed that the decomposition of azopolyimides proceeds in three or four stages, regardless of heating rate.