This paper investigates the notion of approach nearness spaces. Using clusters, completion of an approach nearness space is constructed, which is a unified study of completion in the context of metric spaces, uniform approach spaces, weakly symmetric approach spaces and nearness spaces. Another generalization of completeness, called ultrafilter completeness is introduced to prove the Niemytzki–Tychonoff theorem for approach nearness spaces. Both definitions of completions are shown to be equivalent in a limit-regular approach space. Various examples are given to support the present study.
In the present paper, we have made a category theoretic and lattice theoretic study of some nearness-like structures in the L-approach theory. Using L-grills, the notion of L-approach grill structure is introduced as a characterization of L-approach grill merotopy on X; their categorical perspectives and implications are also investigated. A number of illustrative examples are included.
The objective of the present paper is to develop a comprehensive theory of entropy in the realm of a gs-space involving the notion of a generalized s-map, and to study dynamics of a p-preserving system . It is obtained that the entropy of p-preserving systems is isomorphism invariant, and also entropy of an invertible p-preserving system vanishes provided the corresponding Rokhlin space admits a one-sided generator.
In the present paper, we have studied quantum dynamical systems of difference posets, their equivalence, subsystems and spectra. It is shown that every subsystem of a mixing quantum dynamical system is mixing, and also a bounded spectrum of quantum dynamical systems has its supremum, and all such suprema are equivalent. Entropy of subsystems of a quantum dynamical system of orthomodular lattices is also investigated.
The aim of the present paper is to study extension of functions defined on a subfamily of a D-poset E, as well as ν-measurability of elements in E. We have also considered monotone extensions for a nested sequence of generalized quantum spaces.
The objective of the present paper is to introduced and study the notion of μ-approximation by a subfamily K of a difference poset E. Various properties are proved and then applied to obtain some crucial results including a generalization of the Marczewski Theorem which states that countable compactness is sufficient for σ-additivity of a supermodular measure μ.
In order to identify new solar water splitting photoanodes, Fe–Ti–W–O materials libraries were fabricated by combinatorial reactive co-sputtering and investigated by high-throughput characterization methods to elucidate compositional, thickness, and structural properties. In addition, photoelectrochemical measurements such as potentiodynamic photocurrent determination and open circuit potential measurements were performed using an automated scanning droplet cell. In the thin-film library, a quaternary photoactive region Fe30–49Ti29–55W13–22Ox was identified as a hit composition region, comprising binary and ternary phases. The identified region shows a distinct surface morphology with larger grains (∼200 nm) being embedded into a matrix of smaller grains (∼80–100 nm). A maximum photocurrent density of 117 μA/cm2 at a bias potential of 1.45 V vs. RHE in NaClO4 as an electrolyte under standard solar simulating conditions was recorded. Additional samples with compositions from the hit region were fabricated by reactive co-sputtering and spin coating followed by annealing. Synchrotron X-ray diffraction of sputtered Fe32Ti52W16Ox thin-films, annealed in air (600 °C, 700 °C, 800 °C) revealed the presence of the phases FeTiO3 and Ti0.54W0.46O2. The composition Fe48Ti30W22Ox from the hit region was fabricated by spin coating and subsequent annealing for a detailed investigation of its structure and photoactivity. After annealing the spin-coated sample at 650 °C for 6 h, X-ray diffraction results showed a dominant pattern with narrow diffraction lines belonging to a distorted FeWO4 (ferberite) phase along with broad diffraction lines addressed as Fe2TiO5 and in a small fraction also, Fe1.7Ti0.23O3. In hematite, Fe can be substituted by Ti, therefore we suggest that in the newfound ferberite-type phase, Ti partially substitutes for Fe leading to a small lattice distortion and a doubling of the monoclinic unit cell. In addition, Na from the substrate stabilizes the new phase: its tentative chemical formula is NaxFe0.33Ti0.67W2O8. A maximum photocurrent density of around 0.43 mA/cm2 at 1.45 V vs. RHE in 1M NaOH (pH ∼ 13.6) as an electrolyte was measured. Different aspects of the dependence of annealing and precursor solution concentration on phase transformation and photoactivity are discussed.