Karen Serrano, Margaret Bezrutczyk, Danielle Goudeau
et al.
Abstract The symbiotic interaction of plants with arbuscular mycorrhizal (AM) fungi is ancient and widespread. Plants provide AM fungi with carbon in exchange for nutrients and water, making this interaction a prime target for crop improvement. However, plant–fungal interactions are restricted to a small subset of root cells, precluding the application of most conventional functional genomic techniques to study the molecular bases of these interactions. Here we used single-nucleus and spatial RNA sequencing to explore both Medicago truncatula and Rhizophagus irregularis transcriptomes in AM symbiosis at cellular and spatial resolution. Integrated, spatially registered single-cell maps revealed infected and uninfected plant root cell types. We observed that cortex cells exhibit distinct transcriptome profiles during different stages of colonization by AM fungi, indicating dynamic interplay between both organisms during establishment of the cellular interface enabling successful symbiosis. Our study provides insight into a symbiotic relationship of major agricultural and environmental importance and demonstrates a paradigm combining single-cell and spatial transcriptomics for the analysis of complex organismal interactions.
We study polarization for nonbinary channels with input alphabet of size q = 2r, r = 2,3,.... Using Arikan's polarizing kernel H2, we prove that virtual channels that arise in the process of polarization converge to q-ary channels with capacity 1,2,..., r bits, and that the total transmission rate approaches the symmetric capacity of the channel. This leads to an explicit transmission scheme for q-ary channels. The error probability of decoding using successive cancellation behaves as exp(-Nα), where N is the code length and α is any constant less than 0.5.
Let C be an elliptic curve over Q. Let N be the conductor of C. The Taniyama conjecture asserts that there is a non-constant map of algebraic curves X 0 (N) — C which is defined over Q. Here, X o (N) is the standard modular curve associated with the problem of classifying elliptic curves E together with cyclic subgroups of E having order N.
In this paper we use the Hecke algebra of type B to define a new algebra S which is an analogue of the q‐Schur algebra. We show that S has ‘generic’ basis which is independent of the choice of ring and the parameters q and Q. We then construct Weyl modules for S and obtain, as factor modules, a family of irreducible S‐modules defined over any field. 1991 Mathematics Subject Classification: 16G99, 20C20, 20G05.