Commutative L-algebras and measure theory

Measure and integration theory for finitely additive measures, including vector-valued measures, is shown to be essentially covered by a class of commutative L-algebras, called measurable algebras. The domain and range of any measure is a commutative L-algebra. Each measurable algebra embeds into its structure group, an abelian group with a compatible lattice order, and each (general) measure extends uniquely to a monotone group homomorphism between the structure groups. On the other hand, any measurable algebra X is shown to be the range of an essentially unique measure on a measurable space, which plays the role of a universal covering. Accordingly, we exhibit a fundamental group of X, with stably closed subgroups corresponding to a special class of measures with X as target. All structure groups of measurable algebras arising in a classical context are archimedean. Therefore, they admit a natural embedding into a group of extended real-valued continuous functions on an extremally disconnected compact space, the Stone space of the measurable algebra. Extending Loomis’ integration theory for finitely additive measures, it is proved that, modulo null functions, each integrable function can be represented by a unique continuous function on the Stone space.

Synchronous RNA conformational changes trigger ordered phase transitions in crystals

Time-resolved studies of biomacromolecular crystals have been limited to systems involving only minute conformational changes within the same lattice. Ligand-induced changes greater than several angstroms, however, are likely to result in solid-solid phase transitions, which require a detailed understanding of the mechanistic interplay between conformational and lattice transitions. Here we report the synchronous behavior of the adenine riboswitch aptamer RNA in crystal during ligand-triggered isothermal phase transitions. Direct visualization using polarized video microscopy and atomic force microscopy shows that the RNA molecules undergo cooperative rearrangements that maintain lattice order, whose cell parameters change distinctly as a function of time. The bulk lattice order throughout the transition is further supported by time-resolved diffraction data from crystals using an X-ray free electron laser. The synchronous molecular rearrangements in crystal provide the physical basis for studying large conformational changes using time-resolved crystallography and micro/nanocrystals.

Fuzzy Lattice Order Group Decision for Preference Ranking in Conflict Analysis

Methods of fuzzy multi-objective lattice order decision making (F-MOLODM) for analyses of the fuzzy and intransitive preferences of decision makers involved in a conflict analysis are devised for the preference ranking of the states in a conflict, which defines the trapezoidal fuzzy number of the preference and constructs a multi-objective group decision-making fuzzy preference matrix. An algorithm for F-LOGDM is proposed to capture uncertainty and intransitive decision problems, and the relationship among lattice elements of the preference structure under fuzzy environments is defined. The application of these decision technologies to the square dance conflict illustrate how the method proposed in this paper can be utilized in practice. The performance of the lattice order preference ranking method applied to the conflict is compared with that of traditional methods. Research shows that either lattice order preference ranking, or traditional methods acquires Nash stability. However, the stable state under traditional methods will be reached only if all players pay a certain price.

Self-assembly of highly ordered DNA origami lattices at solid-liquid interfaces by controlling cation binding and exchange

The surface-assisted hierarchical self-assembly of DNA origami lattices represents a versatile and straightforward method for the organization of functional nanoscale objects such as proteins and nanoparticles. Here, we demonstrate that controlling the binding and exchange of different monovalent and divalent cation species at the DNA-mica interface enables the self-assembly of highly ordered DNA origami lattices on mica surfaces. The development of lattice quality and order is quantified by a detailed topological analysis of high-speed atomic force microscopy (HS-AFM) images. We find that lattice formation and quality strongly depend on the monovalent cation species. Na+ is more effective than Li+ and K+ in facilitating the assembly of high-quality DNA origami lattices, because it is replacing the divalent cations at their binding sites in the DNA backbone more efficiently. With regard to divalent cations, Ca2+ can be displaced more easily from the backbone phosphates than Mg2+ and is thus superior in guiding lattice assembly. By independently adjusting incubation time, DNA origami concentration, and cation species, we thus obtain a highly ordered DNA origami lattice with an unprecedented normalized correlation length of 8.2. Beyond the correlation length, we use computer vision algorithms to compute the time course of different topological observables that, overall, demonstrate that replacing MgCl2 by CaCl2 enables the synthesis of DNA origami lattices with drastically increased lattice order.

Influence of Chitosan Addition on Resorcinol–Formaldehyde Xerogel Structure

Gels are usually not environment-friendly due to their difficult biodegradability. Therefore, the addition of chitosan, even in small amounts, will make such gels biodegradable and thus can be useful in many applications that require environment-friendly materials. The addition of small quantities of chitosan to the reacting solution resorcinol–formaldehyde xerogel was investigated. Different hybrid resorcinol–formaldehyde–chitosan xerogels were characterized by different techniques, including Raman spectra, FTIR, XRD, TGA, SEM, surface area and porosity analyzer, and CHNS/O microanalyzer. It was seen that the addition of chitosan, even in a minor quantity, has a significant influence on the structural features of the resulting xerogels. The lattice order and crystallinity, chemical functions, thermal stability, morphology, elemental ratio, pore structure, and appearance were changed by adding chitosan into the xerogel structure.

Broad studies of graphene and low-density polyethylene composites

Graphene (Gr) distribution in low-density polyethylene (LDPE) considerably increased thermal stability, thermal conductivity, mechanical properties, and flexural properties of LDPE/Gr composites. Addition of Grs to LDPE postponed the time for making the polymer brittle. High specific surface area and superior properties of Gr improved thermal stability, conductivity, storage modulus, and mechanical properties of composites. The electrical conductivity of LDPE/Grs composites upgraded owing to the thermal stability of Grs in LDPE matrix. In terms of rheology, the addition of Grs augmented viscosity of the LDPE matrix. Addition of Grs to LDPE nucleates crystallization by reducing the activation energy along with rising crystallization onset temperature. Adding Gr facilitated decreasing aggregation, expanded crystallinity, improved the local lattice order of LDPE/Grs, and advanced Grs contact with LDPE. Thus, on a macroscopic scale, Gr constrains mobility of polymer chains, causing a growth in stiffness and strength of the composite. The distribution of Grs in LDPE at micron size scale was verified by atomic force microscopy and other microscopic testers. With further Grs inclusions to LDPE, the activation energy reduced, Grs proceeded as nucleating agents throughout the crystallization of composites, and increased the enhancement of relative crystallinity of LDPE/Gr compounds. The percolation phenomenon of LDPE/Gr composite occurred about 0.5 wt% of Gr loading. Due to further addition of Gr to LDPE, the impermeability of oxygen through the conduit raised somehow the LDPE/Gr sample with 0.5 wt% Gr content, generated a sharp improvement, and dropped fuel permeation with about 37% in comparison with pure LDPE.