Relative Entropy in CFT

By using Araki's relative entropy, Lieb's convexity and the theory of singular integrals, we compute the mutual information associated with free fermions, and we deduce many results about entropies for chiral CFT's which are embedded into free fermions, and their extensions. Such relative entropies in CFT are here computed explicitly for the first time in a mathematical rigorous way. Our results agree with previous computations by physicists based on heuristic arguments; in addition we uncover a surprising connection with the theory of subfactors, in particular by showing that a certain duality, which is argued to be true on physical grounds, is in fact violated if the global dimension of the conformal net is greater than $1.$Comment: 31 page

Additional file 10: of Insights into the cotton anther development through association analysis of transcriptomic and small RNA sequencing

Figure S3. Target gene enrichment and cluster analysis. A: KEGG enrichment of miRNAs, differentially expressed between SCP and MRP, target genes; B: Expression profiles of miRNAs target genes; C: GO enrichment of target genes (BP: Biological process; CC: Cellular component; MF: Molecular function). (PDF 429 kb

Additional file 3: of Insights into the cotton anther development through association analysis of transcriptomic and small RNA sequencing

Table S2. Differentially expressed genes KEGG enriched TOP10. (XLSX 21 kb

Additional file 13: of Insights into the cotton anther development through association analysis of transcriptomic and small RNA sequencing

Figure S6. The four stages of cotton pollen development. Cotton bud and anther sizes at the sporogenuous cell proliferation (SCP), meiotic phase (MP), microspore release period (MRP) and pollen maturity (PM) four continuous stages. A: Bud; B: Stamen; C: Anther; D: Pollen cellular state under optical microscope (stained with acetocarmine). (PDF 165 kb

Additional file 2: of Insights into the cotton anther development through association analysis of transcriptomic and small RNA sequencing

Figure S1. Sample-to-Sample clustering. (PDF 9 kb

Additional file 9: of Insights into the cotton anther development through association analysis of transcriptomic and small RNA sequencing

Table S7. Prediction of target genes for differentially expressed miRNAs, which between SCP and PM and KEGG enrichment of target genes. (XLSX 11 kb

Additional file 8: of Insights into the cotton anther development through association analysis of transcriptomic and small RNA sequencing

Table S6. Target gene prediction for differentially expressed miRNAs, which between SCP and MRP, and KEGG enrichment of target genes. (XLSX 11 kb

Additional file 6: of Insights into the cotton anther development through association analysis of transcriptomic and small RNA sequencing

Table S4. Differential genes expression. (XLSX 31 kb

Additional file 1: of Insights into the cotton anther development through association analysis of transcriptomic and small RNA sequencing

Table S1. Data statistics for transcriptome sequencing. (XLSX 11 kb

Additional file 12: of Insights into the cotton anther development through association analysis of transcriptomic and small RNA sequencing

Figure S5. Three miRNAs were used for the judgment of anther development stage according to the expression relationship were confirmed by real-time quantitative RT-PCR. The order of expression amount is ghr-miR396a > Dt_chr7_8575 > ghr-miR2948-5p in SCP, Dt_chr7_8575 > ghr-miR396a > ghr-miR2948-5p in MP, ghr-miR2948-5p > Dt_chr7_8575 > ghr-miR396a in MRP, and Dt_chr7_8575 > ghr-miR2948-5p > ghr-miR396a in PM. (PDF 67 kb