A limit field for orthogonal range searches in two-dimensional random point search trees

We consider the cost of general orthogonal range queries in random quadtrees. The cost of a given query is encoded into a (random) function of four variables which characterize the coordinates of two opposite corners of the query rectangle. We prove that, when suitably shifted and rescaled, the random cost function converges uniformly in probability towards a random field that is characterized as the unique solution to a distributional fixed-point equation. We also state similar results for 2-d trees. Our results imply for instance that the worst case query satisfies the same asymptotic estimates as a typical query, and thereby resolve an open question of Chanzy et al. (2001).     

Homology Modeling of Cyt2Ca1 of Bacillus thuringiensis and Its Molecular Docking with Inositol Monophosphate

Cyt2Ca1 is an insecticidal crystal protein produced by Bacillus thuringiensis ET29 during its stationary phase, and this δ‐endotoxin demonstrates remarkable insecticidal activity against not only insects of the order Coleoptera, but also against fleas, and in particular the larvae of the cat flea, Ctenocephalides felis. The first theoretical model of the three‐dimensional structure of Cyt2Ca1 was predicted and compared with Cyt2Aa, which is lethal to insect larvae. The three‐dimensional structure of the Cyt2Ca1 was obtained by homology modeling on the structures of the Cyt2Aa protein. The deduced model resembles previously reported Cyt2Aa toxin. A binding mode of inositol monophosphate as a polar head group of the putative membrane phospholipid ligand to Cyt2Ca1 was presented using molecular docking. The residues of Leu9, Glu21, Tyr23 and Gln110 of the Cyt2Ca1 toxin are responsible for the interactions with inositol monophosphate via eight hydrogen bonds. Those residues could be important for receptor recognition. This binding simulation will be helpful for the design of mutagenesis experiments aimed at the improvement of toxicity, and lead to a deep understanding of the mechanism of action of Cyt toxins.     

Stationary solitons of a three-wave model generated by Type II second-harmonic generation in quadratic media

In this paper, we prove the uniqueness of stationary standing wave solutions of an optical model generated by Type II Second Harmonic Generation (SHG) with behaviors tending to zero at infinity under certain conditions on parameters. In addition, we provide the same issues for the Dirichlet boundary value problems on the ball centered at the origin. A classification of solutions for radial case is also established.     

High school students' use of representations in physics problem solving

Findings from physics education research strongly point to the critical need for teachers’ use of multiple representations in their instructional practices such as pictures, diagrams, written explanations, and mathematical expressions to enhance students' problem‐solving ability. In this study, we explored use of problem‐solving tasks for generating multiple representations as a scaffolding strategy in a high school modeling physics class. Through problem‐solving cognitive interviews with students, we investigated how a group of students responded to the tasks and how their use of such strategies affected their problem‐solving performance and use of representations as compared to students who did not receive explicit, scaffolded guidance to generate representations in solving similar problems. Aggregated data on students' problem‐solving performance and use of representations were collected from a set of 14 mechanics problems and triangulated with cognitive interviews. A higher percentage of students from the scaffolding group constructed visual representations in their problem‐solving solutions, while their use of other representations and problem‐solving performance did not differ with that of the comparison group. In addition, interviews revealed that students did not think that writing down physics concepts was necessary despite being encouraged to do so as a support strategy.     

Two criteria to check whether ideals are direct sums of cyclically presented modules

A famous theorem of commutative algebra due to I. M. Isaacs states that “if every prime ideal of R is principal, then every ideal of R is principal”. Therefore, a natural question of this sort is “whether the same is true if one weakens this condition and studies rings in which ideals are direct sums of cyclically presented modules?” The goal of this paper is to answer this question in the case R is a commutative local ring. We obtain an analogue of Isaacs's theorem. In fact, we give two criteria to check whether every ideal of a commutative local ring R is a direct sum of cyclically presented modules, it suffices to test only the prime ideals or structure of the maximal ideal of R. As a consequence, we obtain: if R is a commutative local ring such that every prime ideal of R is a direct sum of cyclically presented R-modules, then R is a Noetherian ring. Finally, we describe the ideal structure of commutative local rings in which every ideal of R is a direct sum of cyclically presented R-modules.     

On Eisenstein polynomials and zeta polynomials

Eisenstein polynomials, which were defined by Oura, are analogues of the concept of an Eisenstein series. Oura conjectured that there exist some analogous properties between Eisenstein series and Eisenstein polynomials. In this paper, we provide new analogous properties of Eisenstein polynomials and zeta polynomials. These properties are finite analogies of certain properties of Eisenstein series.     

Preparation of diatomite–TiO<Subscript>2</Subscript> composite for photodegradation of bisphenol-A in water

To enhance the photodegradation performance of pure titanium dioxide (TiO₂), diatomite was used as a porous carrier to immobilize TiO₂ powders using calcination method. The photodegradation of bisphenol-A (BPA; 4,4′-isopropylidenediphenol), which has been listed as one of endocrine disrupting chemicals, was carried out in a batch suspension reactor using pure TiO₂ powders and diatomite–TiO₂ composites, respectively. Under the controlled conditions, the photocatalytic efficiencies of the BPA degradation by the diatomite–TiO₂ composites can be found to be higher than those by pure TiO₂ powders. This result should be attributable to the accessibility of the BPA molecules to the surface of TiO₂ particle in the modified photocatalysts, showing that the enrichment of the organic solute enhanced the rate of photodegradation on the diatomite–TiO₂ composite. However, the photodegradation efficiency was not dependent on the pore properties of these TiO₂ photocatalysts. The experimental results further indicated that the photodegradation kinetics for the destruction of BPA in water followed the first-order model well. The apparent first-order reaction constants (k _obs), thus obtained from the fittings of the model, were in line with the destruction-removal efficiencies of BPA in all the photocatalytic experiments.     

A Complete Multiscale Modelling Approach for Polymer–Clay Nanocomposites

We present an innovative, multiscale computational approach to probe the behaviour of polymer–clay nanocomposites (PCNs). Our modeling recipe is based on 1) quantum/force‐field‐based atomistic simulation to derive interaction energies among all system components; 2) mapping of these values onto mesoscopic bead–field (MBF) hybrid‐method parameters; 3) mesoscopic simulations to determine system density distributions and morphologies (i.e., intercalated versus exfoliated); and 4) simulations at finite‐element levels to calculate the relative macroscopic properties. The entire computational procedure has been applied to two well‐known PCN systems, namely Nylon 6/Cloisite 20A and Nylon 6/Cloisite 30B, as test materials, and their mechanical properties were predicted in excellent agreement with the available experimental data. Importantly, our methodology is a truly bottom‐up approach, and no “learning from experiment” was needed in any step of the entire procedure.     

RNA Aptamers That Reversibly Bind Photoresponsive Azobenzene‐Containing Peptides

Modulation of biological networks assembled by diverse interactions among biologically active molecules has provided a platform for innovative biotechnologies. Here, we report RNA aptamers that bind to a photoresponsive peptide (KRAzR; Lys‐Arg‐azobenzene‐Arg) containing azobenzene chromophore, which can change its structure by photoirradiation. Aptamers were identified after 10 cycles of an in vitro selection procedure starting with a DNA library containing a 70 nt random region. Surface plasmon resonance (SPR) analysis demonstrated that interactions between aptamers and KRAzR were fully controlled by appropriate photoirradiation to the SPR sensor chip. Upon irradiation of 360 nm on the KRAzR‐immobilized surface, the binding of each aptamer to the surface was significantly decreased. Subsequent photoirradiation of the same surface with 430 nm restored the aptamer binding to the surface. We also observed that direct photoirradiation of the aptamer–peptide complex on a gold surface actively promoted dissociation of the complex. Furthermore, a doped reselection method was applied to acquire structural and sequence information of aptamer 66. From a data analysis of the conserved region and the mutation frequency, we were able to select a plausible secondary structure among three candidates predicted by computational folding simulation.     

Insights into Structure Direction of Microporous Aluminophosphates: Competition between Organic Molecules and Water

A combination of experimental characterisation techniques and computational modelling has allowed us to gain insight into the molecular features governing structure direction in the synthesis of microporous aluminophosphates. The occlusion of three different structure‐directing agents (SDAs), triethylamine (TEA), benzylpyrrolidine (BP) and (S )‐(?)‐N‐benzylpyrrolidine‐2‐methanol (BPM), within the AFI structure during its crystallisation, together with the simultaneous incorporation of water, has been experimentally measured. We found a higher incorporation of organic molecules in the structure obtained with BPM, while a higher water (and lower organic) content is found for the ones obtained with TEA and BP as SDAs. The computational study provides a thermodynamic explanation for the observed behaviour in terms of the relative stabilisation energy of the SDAs and water molecules within the AFI framework compared with when they are in aqueous solution, and demonstrates that a competition for preferential occupation exists between water and organic SDAs, which is a function of the interaction with the inorganic framework. The lower interaction of TEA and BP molecules with the AFI structure promotes the simultaneous incorporation of water molecules in the 12‐membered‐ring (MR) channel, to increase the host–guest interaction energy and thus the thermodynamic stability. The presence of strongly interacting methanol groups in the BPM molecules leads to the incorporation of only organic molecules within the 12‐MR channels. Our results demonstrate the essential role that water molecules play in the stabilisation of hydrophilic microporous aluminophosphates; a minimum amount of organic SDA is, however, essential for a templating role of the microporous architecture.