Bounding the gap between the McCormick relaxation and the convex hull for bilinear functions

We investigate how well the graph of a bilinear function \(b{:}\;[0,1]^n\rightarrow \mathbb {R}\) can be approximated by its McCormick relaxation. In particular, we are interested in the smallest number c such that the difference between the concave upper bounding and convex lower bounding functions obtained from the McCormick relaxation approach is at most c times the difference between the concave and convex envelopes. Answering a question of Luedtke, Namazifar and Linderoth, we show that this factor c cannot be bounded by a constant independent of n. More precisely, we show that for a random bilinear function b we have asymptotically almost surely \(c\geqslant \sqrt{n}/4\). On the other hand, we prove that \(c\leqslant 600\sqrt{n}\), which improves the linear upper bound proved by Luedtke, Namazifar and Linderoth. In addition, we present an alternative proof for a result of Misener, Smadbeck and Floudas characterizing functions b for which the McCormick relaxation is equal to the convex hull.     

The transport dynamics in complex systems governing by anomalous diffusion modelled with Riesz fractional partial differential equations

In this paper, numerical solutions of fractional Fokker–Planck equations with Riesz space fractional derivatives have been developed. Here, the fractional Fokker–Planck equations have been considered in a finite domain. In order to deal with the Riesz fractional derivative operator, shifted Grünwald approximation and fractional centred difference approaches have been used. The explicit finite difference method and Crank–Nicolson implicit method have been applied to obtain the numerical solutions of fractional diffusion equation and fractional Fokker–Planck equations, respectively. Numerical results are presented to demonstrate the accuracy and effectiveness of the proposed numerical solution techniques. Copyright © 2016 John Wiley & Sons, Ltd.     

Adhesion of nonplanar wrinkled surfaces

Topological patterns on polymer surfaces can significantly alter and control adhesion. In this study, the effect of surface wrinkles on a spherical surface on adhesion has been studied. Surface wrinkling induced by swelling of a crosslinked polydimethylsiloxane elastomer constrained by a stiff, thin surface layer (silicate) is used to produce topographic features of various length scales over a large curved area. By controlling the properties of the stiff layer and the applied strain conditions, surface wrinkles of varying amplitude and wavelength are obtained. The effect of wrinkle morphology on adhesion is quantified, and the results display a transition from enhancement of adhesion to decrease depending upon wrinkle dimensions. A simple phenomenological model is proposed that describes the change of adhesion behavior as a function of wrinkle morphology. Our results provide a critical understanding toward tuning the adhesion behavior of nonplanar surfaces consisting of periodic topographic structures. © 2010 Wiley Periodicals, Inc. J Polym Sci Part B: Polym Phys, 2011     

Syntheses, transfection efficacy and cell toxicity properties of novel cholesterol-based gemini lipids having hydroxyethyl head group

We have synthesized five new cholesterol based gemini cationic lipids possessing hydroxyethyl (-CH(2)CH(2)OH) function on each head group, which differ in the length of the polymethylene spacer chain. These gemini lipids are important for gene delivery processes as they possess pre-optimized molecular features, e.g., cholesterol backbone, ether linkage and a variable spacer chain between both the headgroups of the gemini lipids. Cationic liposomes were prepared from each of these lipids individually and as a mixture of individual cationic gemini lipid and 1,2-dioleoyl phosphatidylethanolamine (DOPE). Each gemini lipid based formulation induced better transfection activity than that of their monomeric counterpart. One such gemini lipid with a -(CH(2))(12)- spacer, HG-12, showed dramatic increase in the mean fluorescence intensity due to the expression of green-fluorescence protein (GFP) in the presence of 10% FBS compared to the conditions where there was no serum. Other gemini lipids retained their gene transfection efficiency without any marked decrease in the presence of serum. The only exception was seen with the gemini with a -(CH(2))(3)- spacer, HG-3, which on gene transfection in the presence of 10% FBS lost ~70% of its transfection efficiency. Overall the gemini lipid with a -(CH(2))(5)- spacer, HG-5, showed the highest transfection activity at N/P (lipid/DNA) ratio of 0.5 and lipid : DOPE molar ratio of 2. Upon comparison of the relevant parameters, e.g., %-transfected cells, the amount of DNA transfected to each cell and %-cell viability all together against Lipofectamine 2000, one of the best commercial transfecting agents, the optimized lipid formulation based on DOPE/HG-5 was found to be comparable. In terms of its ability to induce gene-transfer in the presence of serum and shelf-life DOPE/HG-5 liposome was found to be superior to its commercial counterpart. Confocal imaging analysis confirmed that in the presence of 10% serum using a Lipid : DOPE of 1 : 4 and N/P charge ratio of 0.75 with 1.2 μg DNA per well, HG-5 is better than Lipofectamine 2000.     

Two valued measure and summability of double sequences in asymmetric context

We extend the ideas of convergence and Cauchy condition of double sequences extended by a two valued measure (called ??-statistical convergence/Cauchy condition and convergence/Cauchy condition in ??-density, studied for real numbers in our recent paper [7]) to a very general structure like an asymmetric (quasi) metric space. In this context it should be noted that the above convergence ideas naturally extend the idea of statistical convergence of double sequences studied by Móricz [15] and Mursaleen and Edely [17]. We also apply the same methods to introduce, for the first time, certain ideas of divergence of double sequences in these abstract spaces. The asymmetry (or rather, absence of symmetry) of asymmetric metric spaces not only makes the whole treatment different from the real case [7] but at the same time, like [3], shows that symmetry is not essential for any result of [7] and in certain cases to get the results, we can replace symmetry by a genuinely asymmetric condition called (AMA).     

Continuous flow enzyme-catalyzed polymerization in a microreactor

Enzymes immobilized on solid supports are increasingly used for greener, more sustainable chemical transformation processes. Here, we used microreactors to study enzyme-catalyzed ring-opening polymerization of ε-caprolactone to polycaprolactone. A novel microreactor design enabled us to perform these heterogeneous reactions in continuous mode, in organic media, and at elevated temperatures. Using microreactors, we achieved faster polymerization and higher molecular mass compared to using batch reactors. While this study focused on polymerization reactions, it is evident that similar microreactor based platforms can readily be extended to other enzyme-based systems, for example, high-throughput screening of new enzymes and to precision measurements of new processes where continuous flow mode is preferred. This is the first reported demonstration of a solid supported enzyme-catalyzed polymerization reaction in continuous mode.     

A new numerical approach for single rational soliton solution of Chen‐Lee‐Liu equation with Riesz fractional derivative in optical fibers

In this paper, time‐splitting spectral approximation technique has been proposed for Chen‐Lee‐Liu (CLL) equation involving Riesz fractional derivative. The proposed numerical technique is efficient, unconditionally stable, and of second‐order accuracy in time and of spectral accuracy in space. Moreover, it conserves the total density in the discretized level. In order to examine the results, with the aid of weighted shifted Grünwald‐Letnikov formula for approximating Riesz fractional derivative, Crank‐Nicolson weighted and shifted Grünwald difference (CN‐WSGD) method has been applied for Riesz fractional CLL equation. The comparison of results reveals that the proposed time‐splitting spectral method is very effective and simple for obtaining single soliton numerical solution of Riesz fractional CLL equation.     

Formation of nanoscale aggregates of a coumarin derivative in Langmuir–Blodgett film

In the present communication, we report the formation of organized nanoscale aggregates of a coumarin derivative 7 Hydroxy-N-Octadecyl Coumarin-3-Carboxamide (7HNO3C) at the air–water interface and in Langmuir–Blodgett (LB) films in the presence and absence of stearic acid (SA). A pressure-area isotherm reveals that the 7HNO3C form stable monolayer at the air–water interface. However, the stability can be improved by mixing it with a fatty acid stearic acid (SA). The miscibility study shows that the nature of interaction is strongly dependent on the mixing ratio and surface pressure. At a mole fraction of 0.4 of 7HNO3C in SA, the attractive and repulsive interaction between these two molecules balance each other forming a stable film with nanoscale aggregates. UV-Vis absorption spectroscopic studies reveal the nature of the aggregates in LB films. Scanning electron microscopy gives compelling visual evidence of formation of nanoscale aggregates in the mixed LB films.     

Role of vacancy and metal doping on combustive oxidation of Zr/ZrO2 core-shell particles

We studied self-propagated combustion synthesis of transition-metal-doped tetragonal ZrO₂ (t-ZrO₂) with first principles-based one-dimensional diffusion reaction model. The optimal reaction condition for the combustion process was investigated by calculating energetic stability and surface reactivity of oxygen vacancy defects on (101) surface termination of t-ZrO₂ using first-principles density functional methods. In the first-principles model, the surface was doped with 14 different metal impurities in the 4th and 5th row of the periodic table to examine the role of transition-metal doping on the combustion process. Results indicate that there are clear trends in the defect stability and reactivity depending upon the type of metal impurity and their relative location with respect to the oxygen vacancy. Surface density of states and charge density information also show that there is a trade-off between the vacancy stability and chemical activity of the surface defect states. Based on the thermodynamic information obtained from first principles, we analyze the combustion process of a Zr metal particle by using a one-dimensional diffusion-reaction model. The competition between the vacancy-assisted chemisorption and the vacancy diffusion results in an optimal point for rate of combustion reaction with respect to the vacancy stability. From this, we suggest a plausible screening strategy for metal-doping which can be applied at different temperatures and pressures, as well as with different particle sizes. Our analysis indicates that first-principles calculation provides key information that can be subsequently used for an optimization of the reaction rate for a self-sustained combustion process. An explicit inclusion of rates of defect and ionic transport will be introduced into our model in future work.     

Generalized derivations acting on multilinear polynomials in prime rings

Let R be a noncommutative prime ring of characteristic different from 2 with Utumi quotient ring U and extended centroid C, let F, G and H be three generalized derivations of R, I an ideal of R and f(x₁,..., x _n ) a multilinear polynomial over C which is not central valued on R. If

$$F(f(r))G(f(r)) = H(f(r)^2 )$$

for all r = (r₁,..., r _n ) ∈ I ⁿ , then one of the following conditions holds:

1. (1)
   there exist a ∈ C and b ∈ U such that F(x) = ax, G(x) = xb and H(x) = xab for all x ∈ R
    
2. (2)
   there exist a, b ∈ U such that F(x) = xa, G(x) = bx and H(x) = abx for all x ∈ R, with ab ∈ C
    
3. (3)
   there exist b ∈ C and a ∈ U such that F(x) = ax, G(x) = bx and H(x) = abx for all x ∈ R
    
4. (4)
   f(x₁,..., x _n )² is central valued on R and one of the following conditions holds

   1. (a)
      there exist a, b, p, p’ ∈ U such that F(x) = ax, G(x) = xb and H(x) = px + xp’ for all x ∈ R, with ab = p + p’
       
   2. (b)
      there exist a, b, p, p’ ∈ U such that F(x) = xa, G(x) = bx and H(x) = px + xp’ for all x ∈ R, with p + p’ = ab ∈ C.