For spatially dependent functional data, a generalized Karhunen-Lo\`{e}ve expansion is commonly used to decompose data into an additive form of temporal components and correlated coefficients. This structure provides convenient model investigate the space-time interactions, but may not hold for complex spatio-temporal processes. In this work, we introduce concept weak separability, propose form...

We analyze a system of partial differential equations that model potential mean field game controls, briefly MFGC. Such describes the interaction infinitely many negligible players competing to optimize personal value function depends in aggregate on state and, most notably, control choice all other players. A solution corresponds Nash Equilibrium, strategy for which no one player can improve b...

The hydrogensulfate-sulfates Ln(HSO4)(SO4) (Ln: Sm, Eu, Gd, Tb, Dy) have been crystallized from sulfuric acid and their thermal decomposition behavior has studied. crystal structures for all members of the series were refined X-ray single-crystal diffraction data (for all: Tb(HSO4)(SO4) structure type, P21, Z=2, a≈6.66 Å, b≈6.63 c≈6.82 β≈104.6°). Optical spectra (Sm–Dy) magnetic susceptibilitie...