A Holographic Proof of R\'enyi Entropic Inequalities

We prove R\'enyi entropic inequalities in a holographic setup based on the recent proposal for the holographic formula of R\'enyi entropies when the bulk is stable against any perturbation. Regarding the R\'enyi parameter as an inverse temperature, we reformulate the entropies in analogy with statistical mechanics, which provides us a concise interpretation of the inequalities as the positivities of entropy, energy and heat capacity. This analogy also makes clear a thermodynamic structure in deriving the holographic formula. As a by-product of the proof we obtain a holographic formula to calculate the quantum fluctuation of the modular Hamiltonian. A few examples of the capacity of entanglement are examined in detail.Comment: 29 pages, 1 figure; v3: references added, our assumption for the proof clarifie

Entanglement Entropy of Annulus in Three Dimensions

The entanglement entropy of an annulus is examined in a three-dimensional system with or without a gap. For a free massive scalar field theory, we numerically calculate the mutual information across an annulus. We also study the holographic mutual information in the CGLP background describing a gapped field theory. We discover four types of solutions as the minimal surfaces for the annulus and classify the phase diagrams by varying the inner and outer radii. In both cases, we find the mutual information satisfies the monotonicity dictated by the unitarity and decays exponentially fast as the gap scale is increased. We speculate this is a universal behavior in any gapped system.Comment: 29 pages, 13 figures, v2: references added, minor change

Renormalized Entanglement Entropy on Cylinder

We develop a framework of calculating entanglement entropy for non-conformal field theories with the use of the dilaton effective action. To illustrate it, we locate a theory on a cylinder $\mathbb{R} \times \mathbb{S}^{2}$ and compute entanglement entropy of a cap-like region perturbatively with respect to the mass for a free massive scalar field. A renormalized entanglement entropy (REE) is proposed to regularize the ultraviolet divergence on the cylinder. We find that the REE decreases monotonically both in the small and large mass regions as the mass increases. We confirm all of these behaviors by the numerical calculations, which further shows the monotonic decrease of the REE in the entire renormalization group flow.Comment: 28 pages, 6 figures, v2: a new section on an interesting discrepancy added, some explanations clarifie

Dimension estimate of global attractors for a chemotaxis-growth system and its discretizations

Nakaguchi Etsushi. Dimension estimate of global attractors for a chemotaxis-growth system and its discretizations. 数理解析研究所講究録. No.1693, 2009, p. 143-150

GLOBAL EXISTENCE OF SOLUTIONS TO AN n-DIMENSIONAL PARABOLIC-PARABOLIC SYSTEM FOR CHEMOTAXIS WITH LOGISTIC-TYPE GROWTH AND SUPERLINEAR PRODUCTION

Dedicated to Professor Masayasu Mimura on the occasion of his 75th birthday

Numerical analysis for semilinear evolution equations of parabolic type

AbstractWe study the Galerkin Euler approximations of semilinear evolution equations of parabolic type. We utilize both the semigroup method and the variational method to construct approximate solutions and estimate errors

GLOBAL EXISTENCE OF SOLUTIONS TO AN n-DIMENSIONAL PARABOLIC-PARABOLIC SYSTEM FOR CHEMOTAXIS WITH LOGISTIC-TYPE GROWTH AND SUPERLINEAR PRODUCTION

Dedicated to Professor Masayasu Mimura on the occasion of his 75th birthday