An Introduction to Closed/Open Neighborhood Sums: Minimax, Maximin, and Spread

For a graph G of order |V(G)| = n and a real-valued mapping f:V(G)?\mathbbR{f:V(G)\rightarrow\mathbb{R}}, if S ì V(G){S\subset V(G)} then f(S)=?_{w ? S} f(w){f(S)=\sum_{w\in S} f(w)} is called the weight of S under f. The closed (respectively, open) neighborhood sum of f is the maximum weight of a closed (respectively, open) neighborhood under f, that is, NS[f]=max{f(N[v])|v ? V(G)}{NS[f]={\rm max}\{f(N[v])|v \in V(G)\}} and NS(f)=max{f(N(v))|v ? V(G)}{NS(f)={\rm max}\{f(N(v))|v \in V(G)\}}. The closed (respectively, open) lower neighborhood sum of f is the minimum weight of a closed (respectively, open) neighborhood under f, that is, NS^-[f]=min{f(N[v])|v ? V(G)}{NS^{-}[f]={\rm min}\{f(N[v])|v\in V(G)\}} and NS^-(f)=min{f(N(v))|v ? V(G)}{NS^{-}(f)={\rm min}\{f(N(v))|v\in V(G)\}}. For W ì \mathbbR{W\subset \mathbb{R}}, the closed and open neighborhood sum parameters are NS_W[G]=min{NS[f]|f:V(G)? W{NS_W[G]={\rm min}\{NS[f]|f:V(G)\rightarrow W} is a bijection} and NS_W(G)=min{NS(f)|f:V(G)? W{NS_W(G)={\rm min}\{NS(f)|f:V(G)\rightarrow W} is a bijection}. The lower neighbor sum parameters are NS^-_W[G]=maxNS^-[f]|f:V(G)? W{NS^{-}_W[G]={\rm max}NS^{-}[f]|f:V(G)\rightarrow W} is a bijection} and NS^-_W(G)=maxNS^-(f)|f:V(G)? W{NS^{-}_W(G)={\rm max}NS^{-}(f)|f:V(G)\rightarrow W} is a bijection}. For bijections f:V(G)? {1,2,?,n}{f:V(G)\rightarrow \{1,2,\ldots,n\}} we consider the parameters NS[G], NS(G), NS ⁻[G] and NS ⁻(G), as well as two parameters minimizing the maximum difference in neighborhood sums.     

Assessing the Importance of Cation Size in the Tetragonal-Cubic Phase Transition in Lithium-Garnet Electrolytes**

Lithium garnets are promising solid-state electrolytes for next-generation lithium-ion batteries. These materials have high ionic conductivity, a wide electrochemical window and stability with Li metal. However, lithium garnets have a maximum limit of seven lithium atoms per formula unit (e.g., La₃Zr₂Li₇O₁₂), before the system transitions from a cubic to a tetragonal phase with poor ionic mobility. This arises from full occupation of the Li sites. Hence, the most conductive lithium garnets have Li between 6–6.55 Li per formula unit, which maintains the cubic symmetry and the disordered Li sub-lattice. The tetragonal phase, however, forms the highly conducting cubic phase at higher temperatures, thought to arise from increased cell volume and entropic stabilisation permitting Li disorder. However, little work has been undertaken in understanding the controlling factors of this phase transition, which could enable enhanced dopant strategies to maintain room temperature cubic garnet at higher Li contents. Here, a series of nine tetragonal garnets were synthesised and analysed by variable temperature XRD to understand the dependence of site substitution on the phase transition temperature. Interestingly the octahedral site cation radius was identified as the key parameter for the transition temperature with larger or smaller dopants altering the transition temperature noticeably. A site substitution was, however, found to make little difference irrespective of significant changes to cell volume.     

Machine learning accelerated high-throughput screening of zeolites for the selective adsorption of xylene isomers

The production of widely used polymers such as polyester currently relies upon the chemical separation of and transformation of xylene isomers. The least valuable but most prevalent isomer is meta-xylene which can be selectively transformed into the more useful and expensive para-xylene isomer using a zeolite catalyst but at a high energy cost. In this work, high-throughput screening of existing and hypothetical zeolite databases containing more than two million structures was performed, using a combination of classical simulation and deep neural network methods to identify promising materials for selective adsorption of meta-xylene. Novel anomaly detection techniques were applied to the heavily biased classification task of identifying structures with a selectivity greater than that of the best performing existing zeolite, ZSM-5 (MFI topology). Eight hypothetical zeolite topologies are found to be several orders of magnitude more selective towards meta-xylene than ZSM-5 which may provide an impetus for synthetic efforts to realise these promising materials. Moreover, the leading hypothetical frameworks identified from the screening procedure require a markedly lower operating temperature to achieve the diffusion seen in existing materials, suggesting significant energetic savings if the frameworks can be realised.

A combination of machine learning and high throughput simulation has identified several potential zeolite structures that appear to outperform the leading commercially used material and explained the key factors for high selectivity.     

The diffusion coefficient of a polymer in an array of obstacles is a non-monotonic function of the degree of disorder in the medium

Using Monte Carlo simulations, we have found that there is a surprising non-monotonic dependence of a polymer's diffusion coefficient upon the degree of disorder of the surrounding environment. Starting with a two-dimensional periodic lattice of obstacles, we randomly displace obstacles to create a quenched gel system with a tunable degree of disorder. Very small displacements increase the diffusion coefficient of polymers since they increase the width of the tube through which the polymer chains reptate. As we displace the obstacles further, however, entropic trapping is observed and the diffusion coefficient of the polymer decreases dramatically. This is a striking example of the delicate balance between entropic and frictional effects for a polymer diffusing in a dense system.     

Convex labelings of trees

A convex labeling of a tree T of order n is a one-to-one function f from the vertex set of T into the nonnegative integers, so that f(y) ? (f(x) + f(z))/2 for every path x, y, z of length 2 in T. If, in addition, f(v) ? n ? 1 for every vertex v of T, then f is a perfect convex labeling and T is called a perfectly convex tree. Jamison introduced this concept and conjectured that every tree is perfectly convex. We show that there exists an infinite class of trees, none of which is perfectly convex, and in fact prove that for every n there exists a tree of order n which requires a convex labeling with maximum value at least 6n/5 – 22. We also prove that every tree of order n admits a convex labeling with maximum label no more than n²/8 + 2. In addition, we present some constructive methods for obtaining perfect convex labelings of large classes of trees.     

XYZH systems as potential 1,3-dipoles. Part 62: 1,3-Dipolar cycloaddition reactions of metallo-azomethine ylides derived from α-iminophosphonates

Metallo-azomethine ylides, generated from iminophosphonates in combination with LiBr or AgOAc and bases Et₃N, DBU, t-butyl tetramethylguanidine(BTMG) undergo cycloaddition to give dialkyl pyrrolidine-2-phosphonates along with the corresponding Michael adduct in some cases. Cycloadditions with the chiral dipolarophile 5R-(1′R,2′S,5′R-menthyloxy)-2-(5H)-furanone (MOF) afforded enantiopure cycloadducts.     

Enhanced cellulase production byAspergillus fumigatus fresenius

The compost isolate,Aspergillus fumigatus, produces the exoglucanase, endoglucanase, and Β-glucosidase enzymes required for the breakdown of crystalline cellulose. Cellulose breakdown and extracellular enzyme levels in liquid culture can be affected by low pH values attained during fungal growth. During growth ofA. fumigatus on modified Czapeck-Dox medium containing 1% (w/v) Avicel, it was found that Β-glucosidase activity was lost and endoglucanase activity, reduced, when pH values fell below 3. The effect of buffering (0.2M phosphate, pH 6.15) was examined and compared with the unbuffered medium. Beta-glucosidase activity could be detected throughout the incubation period in the buffered medium and endoglucanase activity was approximately tenfold greater. Exoglucanase activity also showed an increase in the buffered system. Concentrations of phosphate buffer ranging from 0.05 to 0.8M were incorporated into the medium and optimum cellulose breakdown and extracellular enzyme production occurred between 0.1 and 0.2M. Reports suggest that increasing substrate concentration does not improve upon the levels of extracellular cellulase produced because of enzyme inactivation resulting from rapid decreases in pH. Using the buffered medium described previously,A. fumigatus was grown on concentrations of Avicel ranging from 0.5 to 10% (w/v). Cellulose breakdown and extracellular enzyme production was compared with that achieved by a similar nonbuffered system. Endoglucanase and Β-glucosidase activity increased with time and with substrate concentration up to 5% (w/v) in the buffered medium. Beta-glucosidase was negligible at all concentrations of Avicel in the unbuffered medium and endoglucanase activity decreased with increasing substrate concentration with maximum levels approximately eightfold lower than in the buffered system. Extracellular exoglucanase activity was lower in the buffered medium and only increased to levels comparable with those achieved by the unbuffered medium towards the end of the incubation period. In the unbuffered system, exoglucanase activity decreased with increasing substrate concentration, but no such effect was observed in the buffered medium. Negligible growth occurred in both media at 10% (w/v) substrate. The percentage weight loss recorded in the Czapeck-Dox medium also decreased with increasing substrate concentration, while in the buffered medium, over 95% weight loss was recorded in up to 5% (w/v) Avicel. It appeared that cellulose breakdown was more rapid in the buffered medium and a time-course carried out to determine the rate of cellulolysis showed 97% cellulose breakdown after 12 d, corresponding to a plateau and a subsequent decrease in extracellular cellulase levels.     

Two Models for K-12 Hypermediated Earth System Science Lessons Based on Internet Resources

An immense collection of daily and archived earth and space science digital images and database information sources are readily accessible to K-12 teachers through the Internet and the World Wide Web. However, these abundant electronic resources are designed for scientists and often need to be modified to benefit students. Advances in hypermedia as an instructional tool for earth system science education provide a mechanism for designing user-friendly and structured classroom units based on this available data. Two models proposed as a foundation for materials development are the Learning Cycle Model and the Investigation/Experimentation Model. A high demand for Internet-based curricular materials development to take advantage of the current technology suggests an educational research agenda to determine which lesson formats are most effective for learners.     

The Properties of a cellulolytic microbial community growing on pure cellulose and lignocellulose

A microbial community capable of degrading pure cellulose has been isolated from soil using a continuous flow chemostat. The component organisms have been identified as Penicillium nigricans, Paecilomyces liliacinus, Fusarium oxysporum (3 strains), Aspergillus fumigatus, Gliocladium roseum, and Penicillium simplicissimum. Extracellular enzyme production (exoglucanase, endoglucanase, and (Β-glucosidase) and cellulose breakdown by the whole community and the component organisms in pure culture was measured during growth on pure cellulose in batch culture. After 30 d, 50–60% degradation was achieved by Penicillium simplicissimum, Aspergillus fumigatus, and the mixed culture. These cultures also produced the highest levels of extracellular endo- and exoglucanase and were able to produce the full enzyme complex necessary for the degradation of crystalline cellulose. The other isolates were capable of 0–20% degradation and produced lower levels of enzyme activity. Most were unable to produce the full enzyme complex. The results indicated that the whole community was no better at degrading pure cellulose than two members in pure culture. Since the full potential of the community may not be expressed on pure cellulose, breakdown and enzyme production was investigated during growth on lignocellulose. A large proportion of lignocellulosic material consists of xylan and enzymes capable of degrading this component were investigated. The highest percentage degradation and enzyme production (with the exception of Β-glucosidase) after 15 d growth on straw was achieved by the same three cultures. Studies of their growth on hay and straw were extended over a 60-d period. The rate and extent of degradation of hay by P. simplicissimum and A. fumigatus was similar to that of the mixed culture, more than 40% degradation occurring in 40 d. However, differences in levels of enzyme activity were observed. The mixed culture produced lower levels of enzyme activity (with the exception of Β-glucosidase) particularly during the initial states of degradation. Differences in enzyme production by A. fumigatus and P. simplicissimum were also observed. A. fumigatus produced high levels of xylanase and endoglucanase while P. simplicissimum produced high levels of exoglucanase. Similar results were observed on straw. The composition of the lignocellulosic material was measured by sequential chemical extraction of the solubles, hemicellulose, cellulose, and lignin. Changes in the composition during degradation were investigated. During the growth of all three cultures on hay, the total percentage of hemicellulose and cellulose decreased from 84% to less than 45% in 40 d. Despite significant differences in the enzyme activity, P. simplicissimum, A. fumigatus and the mixed culture were capable of degrading lignocellulosic material at the same rate. The production of a full cellulase complex appears to be more important than the production of high levels of individual enzymes.