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Kernel-based learning algorithms have been extensively studied over the past two decades for their successful applications in scientific research and industrial problem-solving. In classical kernel methods, such as kernel ridge regression and support vector machines, an unregularized offset term naturally appears. While its importance can be defended in some situations, it is arguable in others. However, it is commonly agreed that the offset term introduces essential challenges to the optimization and theoretical analysis of the algorithms. In this paper, we demonstrate that Kernel Ridge Regression (KRR) with an offset is closely connected to regularization schemes involving centered reproducing kernels. With the aid of this connection and the theory of centered reproducing kernels, we will establish generalization error bounds for KRR with an offset. These bounds indicate that the algorithm can achieve minimax optimal rates. 

Polyelectrolyte adsorption to surfaces is widely employed in water treatment and mining. However, little is known of the relative interaction strengths between surfaces and polymer. This fundamental property is assumed to be dominated by electrostatics, i.e. attractive interactions between opposite charges, which are set by the overall ionic strength (“salt concentration”) of the solution, and charge densities of the surface and the polymer. A common, counterintuitive, finding is a range of salt concentration over which the amount of adsorbed polyelectrolyte increases as electrostatic interactions are tempered by the addition of salt. After an adsorption maximum, higher salt concentrations then produce the expected gradual desorption of polyelectrolyte. In this work, the salt response of the adsorption of the same narrow molecular weight distribution polycation, poly(N-methyl-4-vinyl pyridinium), PM4VP, to a variety of surfaces was explored. Oxide powders for adsorption included Al2O3, SiO2, Fe2O3, Fe3O4, TiO2, ZnO and CuO. Planar surfaces included silicon wafer, mica, calcium carbonate and CaF2 single crystals. The PM4VP was radiolabeled with 14C so that sensitive, sub-monolayer amounts could be detected. The position of the peak maximum, or the lack of a peak, in response to added salt was used to rank the electrostatic component of the interaction. The importance of charge regulation, a shift in the surface pKa in response to solution species, was highlighted as a mechanism for adsorption on the “wrong” side of the isoelectric point, and also as a factor contributing to the difficulty of reaching the totally desorbed state even at the highest salt concentrations. 

Earth’s inner core is predominantly composed of solid iron (Fe) and displays intriguing properties such as strong shear softening and an ultrahigh Poisson’s ratio. Insofar, physical mechanisms to explain these features coherently remain highly debated. Here, we have studied longitudinal and shear wave velocities of hcp-Fe (hexagonal close-packed iron) at relevant pressure–temperature conditions of the inner core using in situ shock experiments and machine learning molecular dynamics (MLMD) simulations. Our results demonstrate that the shear wave velocity of hcp-Fe along the Hugoniot in the premelting condition, defined asT/T_m(T_m: melting temperature of iron) above 0.96, is significantly reduced by ~30%, while Poisson’s ratio jumps to approximately 0.44. MLMD simulations at 230 to 330 GPa indicate that collective motion with fast diffusive atomic migration occurs in premelting hcp-Fe primarily along [100] or [010] crystallographic direction, contributing to its elastic softening and enhanced Poisson’s ratio. Our study reveals that hcp-Fe atoms can diffusively migrate to neighboring positions, forming open-loop and close-loop clusters in the inner core conditions. Hcp-Fe with collective motion at the inner core conditions is thus not an ideal solid previously believed. The premelting hcp-Fe with collective motion behaves like an extremely soft solid with an ultralow shear modulus and an ultrahigh Poisson’s ratio that are consistent with seismic observations of the region. Our findings indicate that premelting hcp-Fe with fast diffusive motion represents the underlying physical mechanism to help explain the unique seismic and geodynamic features of the inner core. 

Abstract Species functional traits can influence pathogen transmission processes, and consequently affect species' host status, pathogen diversity, and community‐level infection risk. We here investigated, for 143 European waterbird species, effects of functional traits on host status and pathogen diversity (subtype richness) for avian influenza virus at species level. We then explored the association between functional diversity and HPAI H5Nx occurrence at the community level for 2016/17 and 2021/22 epidemics in Europe. We found that both host status and subtype richness were shaped by several traits, such as diet guild and dispersal ability, and that the community‐weighted means of these traits were also correlated with community‐level risk of H5Nx occurrence. Moreover, functional divergence was negatively associated with H5Nx occurrence, indicating that functional diversity can reduce infection risk. Our findings highlight the value of integrating trait‐based ecology into the framework of diversity–disease relationship, and provide new insights for HPAI prediction and prevention.