Abstract. The diametral dimension is an important topological invariant in the category of Fréchet spaces which has been used, e.g., to distinguish types of Stein manifolds. We introduce variants of the classical definition in order to solve an old conjecture of Bessaga, Mityagin, Pe lczyński, and Rolewicz at least for nuclear Fréchet spaces. Moreover, we clarify the relation between an invariant recently introduced by Terzioǧlu and the by now classical condition (Ω) of Vogt and Wagner. Kolmogorov widths and diametral dimensionsKolmogov widths (or diameters) are a quantitative measure for compactness in normed spaces: for absolutely convex sets V and U of a vector space X (typically U is the unit ball of a given norm) and n ∈ N 0 the n-th width is(the dependence on X is notationally surpressed). If V is bounded with respect to U (i.e., δ 0 (V, U ) < ∞) then V is precompact with respect to the the Minkowski functional of U (which is a seminorm with unit ball U ) if and only if δ n (V, U ) → 0. This elementary fact is proposition 1.2 in Pinkus' book [Pin85] where much more information about n-widths can be found.For a locally convex space (l.c.s) X with the system U 0 (X) of absolutely convex 0-neighbourhoods the diametral dimension of X is the sequence spaceThis space is a topological invariant, i.e., if X and Y are isomorphic l.c.s. thenMoreover, X is a Schwartz space (i.e., every 0-neighbourhood U contains another one which is precompact with respect to U ) if and only if ℓ ∞ ⊆ ∆(X). There are several versions of diametral dimensions of locally convex spaces in the literature, all going back to an idea of Pe lczyński [Pe l57] from 1957. The formulation above can be found in [Mit61] where Mityagin refers to a joint work with Bessaga, Pe lczyński, and Rolewicz which, to our best knowledge, eventually did not appear in print.
We report the generation and analysis of single-cell RNA-Seq data (> 38,000 cells) from mouse native retinae and induced pluripotent stem cell (iPSC)-derived retinal organoids at four matched stages of development spanning the emergence of the major retinal cell types. We combine information from temporal sampling, visualization of 3D UMAP manifolds, pseudo-time and RNA velocity analyses, to show that iPSC-derived 3D retinal organoids broadly recapitulate the native developmental trajectories. However, we observe relaxation of spatial and temporal transcriptome control, premature emergence and dominance of photoreceptor precursor cells, and susceptibility of dynamically regulated pathways and transcription factors to culture conditions in retinal organoids. We demonstrate that genes causing human retinopathies are enriched in cell-type specifying genes and identify a subset of disease-causing genes with expression profiles that are highly conserved between human retinae and murine retinal organoids. This study provides a resource to the community that will be useful to assess and further improve protocols for ex vivo recapitulation and study of retinal development.
17We herein report the generation and analysis of single-cell RNA Seq data (> 38,000 cells) from 18 native and iPSC-derived murine retina at four matched developmental stages spanning the 19 emergence of the major retinal cell types. We combine information from temporal sampling, 20 visualization of 3D UMAP manifolds in virtual reality, and RNA velocity analysis to show that 21 iPSC-derived 3D retinal aggregates broadly recapitulate the native developmental trajectories 22 with evidence supporting re-specification from amacrine cells to horizontal and 23 photoreceptor precursor cells, as well as a direct differentiation of Tbr1 + retinal ganglion cells 24 from neuro-epithelium cells. We show relaxation of spatial and temporal transcriptome 25 control, premature emergence and dominance of photoreceptor precursor cells, and 26 susceptibility of dynamically regulated pathways and transcription factors to culture 27 conditions in iPSC-derived retina. We generate bulk ATAC-Seq data for native and iPSC-28 derived murine retina at three of the four same developmental stages identifying ~125,000 29 peaks. The number of ATAC-Seq peaks increased with developmental stage in native but not 30 in iPSC-derived retina. We combine single-cell RNA Seq with ATAC-Seq information and 31demonstrate that approximately halve of the transcription factors that are dynamically 32 Georges et al.Page 2 of 46 regulated during retinal development act as repressors rather than activators. We provide 33 evidence that sets of activators and repressors with cell-type specific expression control 34
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