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

RGR as a function of <i>P</i>_sat and <i>P</i>_sat as a function of <i>G</i>_s in <i>Ficus</i> saplings under two soil nutrient levels.

<p>Significant differences between treatments are marked with different letters (mean ± SE, n = 3, <i>P</i><0.05). Black shapes, fertile, white shapes, infertile. RGR relative growth rate (mg·g⁻¹·d⁻¹); <i>P</i>_sat, light saturated photosynthetic rate (μmol m⁻²s⁻¹); <i>G</i>_s, stomatal conductance (mol H₂O m⁻²s⁻¹).</p

Effect of treatments on root/shoot ratio in <i>Ficus</i> saplings under two soil nutrient levels.

<p>Significant differences between treatments are marked with different letters (mean ± SE, n = 5 in 2009, n = 12 in 2011, <i>P</i><0.05).</p

Effect of treatments on photosynthetic characteristics of <i>Ficus</i> saplings under two soil nutrient levels.

<p>Significant differences between treatments are marked with different letters (mean ± SE, n = 3, <i>P</i><0.05). <i>P</i>_sat, light saturated photosynthetic rate (μmol m⁻²s⁻¹); <i>G</i>_s, stomatal conductance (mol H₂O m⁻²s⁻¹); WUE, water use efficiency (μmol mol⁻¹).</p

Effect of treatments on RGR of <i>Ficus</i> saplings under two soil nutrient levels.

<p>Significant differences between treatments are marked with different letters (mean ± SE, n = 5 in 2009, n = 12 in 2011, <i>P</i><0.05). RGR, relative growth rate (mg·g⁻¹·d⁻¹).</p

Net spillover index of international crude oil prices, onshore RMB exchange rate and offshore RMB exchange rate (%).

Net spillover index of international crude oil prices, onshore RMB exchange rate and offshore RMB exchange rate (%).</p

Directional spillover index of international crude oil prices, onshore RMB exchange rate and offshore RMB exchange rate (%).

Directional spillover index of international crude oil prices, onshore RMB exchange rate and offshore RMB exchange rate (%).</p

Static spillover effect of international crude oil prices, onshore RMB exchange rate and offshore RMB Exchange rate (%).

Static spillover effect of international crude oil prices, onshore RMB exchange rate and offshore RMB Exchange rate (%).</p

Robustness test results.

International crude oil prices has become one of the important factors that affect exchange rate fluctuation. By combining TVP-VAR model and DY time-varying spillover index, this paper measures the two-way fluctuation spillover effect between international crude oil prices and RMB exchange rate from 2011 to 2022. The research results show that: (1) There are obvious time-varying characteristics among international crude oil prices, onshore RMB exchange rate and offshore RMB exchange rate, and they are strengthened when major events occur at home and abroad; (2) The spillover and absorption level among international crude oil prices, onshore RMB exchange rate and offshore RMB exchange rate is relatively limited; (3) After the “exchange rate reform on August 11”, the market participation of RMB exchange rate has increased year by year, and the offshore RMB is mainly net spillover, while the onshore RMB is mainly reflected as a recipient; (4) The offshore RMB exchange rate has a driving effect on the onshore RMB exchange rate. Based on this, it is necessary to orderly promote the development and utilization of new energy, actively promote the market-oriented reform of RMB exchange rate, play the leading role of offshore market, and accelerate the construction of crude oil-RMB settlement system.</div

Compensatory growth of three <i>Ficus</i> species in response to herbivore treatment.

<p>Plant biomass above the line indicates overcompensatory regrowth, on the line indicates fullcompensatory regrowth, below the line indicates undercompensatory regrowth (Biomass in the damaged state  =  biomass in control state, slope  = 1) (n = 5 in 2009, n = 12 in 2011).</p