Foundations

Mobi spaces provide an algebraic abstraction of interpolation between points. In this paper we investigate their relation with geodesic structures. We present a construction of mobi spaces from a parametrized map, formulate conditions under which the mobi axioms follow from a given parametrization, and show that a broad class of second-order differential equations naturally gives rise to mobi spaces. This establishes a general framework linking geodesic motion and mobi structures. As an illustration, we obtain an explicit mobi space associated with a metric of non-constant negative curvature. Taken together, these results suggest that mobi spaces may provide a useful axiomatic framework for the study of geodesic interpolation.

Decomposition integrals on finite spaces have a natural linear programming representation, but their evaluation may require exponentially many variables indexed by subsets of the underlying space. Restricting the admissible sets or collections produces computationally smaller lower approximations and raises the problem of quantifying the resulting loss. We develop an approximation theory for this reduction. For a retained collection and a removed set, the central quantity is the retained collection integral evaluated at the indicator of the removed set; its positive shortfall from the monotone measure value defines the reconstruction defect. We aggregate these local defects through a residual packing problem and obtain simultaneous deletion bounds. For a single removed set, the estimate is sharp and equals the worst-case error on the unit cube. We also determine the smallest uniform multiplicative approximation factor, derive a posteriori certificates from dual reduced profits, and characterize pointwise exactness. The results extend to comparisons of arbitrary collections and to decomposition systems, including deletion of whole member collections. For cardinality-based monotone measures, reconstruction values and approximation factors admit closed forms. These results provide rigorous certificates for sparse evaluation while preserving the underlying non-additive structure.

Foodborne bacterial infections remain an important public health concern, creating a need for effective antimicrobial materials produced through safer and more sustainable methods. Silver nanoparticles (AgNPs) are promising candidates because of their broad antimicrobial activity; however, conventional synthesis commonly involves multistep procedures and potentially hazardous reducing or stabilizing agents. This study presents a rapid, single-step synthesis of AgNPs using D-fructose as a carbohydrate-based reducing and stabilizing agent. The nanoparticles were produced within 3 min by heating an aqueous mixture of AgNO₃ and D-fructose under controlled conditions. UV–Vis spectroscopy showed a surface plasmon resonance band centered at 427 nm, consistent with AgNP formation. X-ray diffraction revealed the characteristic (111), (200), (220), and (311) reflections of face-centered cubic metallic silver, with an estimated crystallite size of 24.7 ± 6.3 nm. Scanning electron microscopy showed nanometric structures with predominantly rounded and irregular morphologies organized into larger aggregates, while energy-dispersive X-ray spectroscopy confirmed Ag as the principal component of the analyzed material (92.92 mass% and 63.61 atom%). Carbon and oxygen were also detected, consistent with the presence of fructose-derived organic species. FTIR spectroscopy further supported this interpretation through the retention and modification of O–H- and C–O-associated bands in the AgNP spectrum. The synthesized AgNPs exhibited bactericidal activity against Escherichia coli ATCC 25922 and Salmonella typhimurium ATCC 14028. MIC/MBC values were 375/750 ppm for E. coli and 4000/5500 ppm for S. typhimurium, demonstrating greater susceptibility of E. coli under the tested conditions. These findings demonstrate a rapid, aqueous, fructose-mediated route for producing crystalline and silver-rich nanostructures with antimicrobial activity, warranting further studies of coating performance, silver migration, activity in food matrices, and cytotoxicity before any food-contact application can be considered.