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We develop information geometric techniques to understand the representations learned by deep networks when they are trained on different tasks using supervised, meta-, semi-supervised and con- trastive learning. We shed light on the following phenomena that relate to the structure of the space of tasks: (1) the manifold of probabilistic models trained on different tasks using different represen- tation learning methods is effectively low-dimen- sional; (2) supervised learning on one task results in a surprising amount of progress even on seem- ingly dissimilar tasks; progress on other tasks is larger if the training task has diverse classes; (3) the structure of the space of tasks indicated by our analysis is consistent with parts of the Word- net phylogenetic tree; (4) episodic meta-learning algorithms and supervised learning traverse differ- ent trajectories during training but they fit similar models eventually; (5) contrastive and semi-su- pervised learning methods traverse trajectories similar to those of supervised learning. We use classification tasks constructed from the CIFAR- 10 and Imagenet datasets to study these phenom- ena. Code is available at https://github.com/grasp- lyrl/picture of space of tasks. 

Macrocyclic poly(glycidyl phenyl ether) (pGPE) synthesized via zwitterionic ring opening polymerization is typically contaminated by chains with linear and tadpole architecture. Although mass spectrometry (MS) analysis can readily confirm the presence of the linear byproduct, due to its unique mass, it is unable to differentiate between the cyclic and tadpole structures, which are constitutional isomers produced by backbiting reactions in monomeric or dimeric chains, respectively. To overcome this problem, ultraperformance reversed-phase liquid chromatography interfaced with electrospray ionization tandem mass spectrometry (UPLC-ESI-MS/MS) was employed. The separation achieved by UPLC revealed that the tadpole isomer elutes before the cyclic structure because of the increased polarity afforded by its distinctive substituents. The ratio of tadpole to cyclic species increased with the degree of polymerization, in agreement with the synthetic method used, as the potential for forming tadpole structures by backbiting is entropically favored in longer polymer chains. Once separated, the two isomers could be independently characterized by tandem mass spectrometry. The macrocyclic and tadpole species exhibit unique fragmentation patterns, including structurally diagnostic fragments for each structure. 

Tandem mass spectrometry (MS2) has been employed to elucidate the topology and branching architecture of star-branched polyethers. The polymers were ionized by matrix-assisted laser desorption/ionization (MALDI) to positive ions and dissociated after leaving the ion source via laser-induced fragmentation. The bond scissions caused under MALDI-MS2 conditions occur preferentially near the core-branch joining points due to energetically favorable homolytic and heterolytic bond cleavages near the core and release of steric strain and/or reduction of crowding. This unique fragmentation mode detaches complete arms from the core generating fragment ion series at the expected molecular weight of each branch. The number of fragment ion distributions observed combined with their mass-to-charge ratios permit conclusive determination of the degree of branching and the corresponding branch lengths, as demonstrated for differently branched homo- and mikto-arm polyether stars synthesized via azide-alkyne click chemistry. The results of this study underscore the utility of MS2 for the characterization of branching architecture and branch lengths of (co)polymers with two or more linear chains attached to a functionalized central core. 

Abstract Self‐assembled nanostructures of rod‐like molecules are commonly limited to nematic or layered smectic structures dominated by the parallel arrangement of the rod‐like components. Distinct self‐assembly behavior of four categories of dendritic rods constructed by placing a tri(hydroxy) group at the apex of dendritic oligo‐fluorenes is observed. Designed hydrogen bonding and dendritic architecture break the parallel arrangement of the rods, resulting in molecules with specific (fan‐like or cone‐like) shapes. While the fan‐shaped molecules tend to form hexagonal packing cylindrical phases, the cone‐shaped molecules could form spherical motifs to pack into various ordered structures, including the Frank–Kasper A15 phase and dodecagonal quasicrystal. This study provides a model system to engineer diverse supramolecular structures by rod‐like molecules and sheds new light into the mechanisms of the formation of unconventional spherical packing structures in soft matter.