Entropy for gravitational Chern-Simons terms by squashed cone method

In this paper we investigate the entropy of gravitational Chern-Simons terms for the horizon with non-vanishing extrinsic curvatures, or the holographic entanglement entropy for arbitrary entangling surface. In 3D we find no anomaly of entropy appears. But the squashed cone method can not be used directly to get the correct result. For higher dimensions the anomaly of entropy would appear, still, we can not use the squashed cone method directly. That is becasuse the Chern-Simons action is not gauge invariant. To get a reasonable result we suggest two methods. One is by adding a boundary term to recover the gauge invariance. This boundary term can be derived from the variation of the Chern-Simons action. The other one is by using the Chern-Simons relation $d\bm{\Omega_{4n-1}}=tr(\bm{R}^{2n})$. We notice that the entropy of $tr(\bm{R}^{2n})$ is a total derivative locally, i.e. $S=d s_{CS}$. We propose to identify $s_{CS}$ with the entropy of gravitational Chern-Simons terms $\Omega_{4n-1}$. In the first method we could get the correct result for Wald entropy in arbitrary dimension. In the second approach, in addition to Wald entropy, we can also obtain the anomaly of entropy with non-zero extrinsic curvatures. Our results imply that the entropy of a topological invariant, such as the Pontryagin term $tr(\bm{R}^{2n})$ and the Euler density, is a topological invariant on the entangling surface.Comment: 19 pag

Holographic Entanglement Entropy for the Most General Higher Derivative Gravity

The holographic entanglement entropy for the most general higher derivative gravity is investigated. We find a new type of Wald entropy, which appears on entangling surface without the rotational symmetry and reduces to usual Wald entropy on Killing horizon. Furthermore, we obtain a formal formula of HEE for the most general higher derivative gravity and work it out exactly for some squashed cones. As an important application, we derive HEE for gravitational action with one derivative of the curvature when the extrinsic curvature vanishes. We also study some toy models with non-zero extrinsic curvature. We prove that our formula yields the correct universal term of entanglement entropy for 4d CFTs. Furthermore, we solve the puzzle raised by Hung, Myers and Smolkin that the logarithmic term of entanglement entropy derived from Weyl anomaly of CFTs does not match the holographic result even if the extrinsic curvature vanishes. We find that such mismatch comes from the `anomaly of entropy' of the derivative of curvature. After considering such contributions carefully, we resolve the puzzle successfully. In general, we need to fix the splitting problem for the conical metrics in order to derive the holographic entanglement entropy. We find that, at least for Einstein gravity, the splitting problem can be fixed by using equations of motion. How to derive the splittings for higher derivative gravity is a non-trivial and open question. For simplicity, we ignore the splitting problem in this paper and find that it does not affect our main results.Comment: 28 pages, no figures, published in JHE

The domination number and the least QQ-eigenvalue

A vertex set $D$ of a graph $G$ is said to be a dominating set if every vertex of $V(G)\setminus D$ is adjacent to at least a vertex in $D$, and the domination number $\gamma(G)$ ($\gamma$, for short) is the minimum cardinality of all dominating sets of $G$. For a graph, the least $Q$-eigenvalue is the least eigenvalue of its signless Laplacian matrix. In this paper, for a nonbipartite graph with both order $n$ and domination number $\gamma$, we show that $n\geq 3\gamma-1$, and show that it contains a unicyclic spanning subgraph with the same domination number $\gamma$. By investigating the relation between the domination number and the least $Q$-eigenvalue of a graph, we minimize the least $Q$-eigenvalue among all the nonbipartite graphs with given domination number.Comment: 13 pages, 3 figure

Brainstem glucose metabolism predicts reward dependence scores in treatment-resistant major depression

BACKGROUND: It has been suggested that individual differences in temperament could be involved in the (non-)response to antidepressant (AD) treatment. However, how neurobiological processes such as brain glucose metabolism may relate to personality features in the treatment-resistant depressed (TRD) state remains largely unclear. METHODS: To examine how brainstem metabolism in the TRD state may predict Cloninger's temperament dimensions Harm Avoidance (HA), Novelty Seeking (NS), and Reward Dependence (RD), we collected (18)fluorodeoxyglucose positron emission tomography ((18)FDG PET) scans in 40 AD-free TRD patients. All participants were assessed with the Temperament and Character Inventory (TCI). We applied a multiple kernel learning (MKL) regression to predict the HA, NS, and RD from brainstem metabolic activity, the origin of respectively serotonergic, dopaminergic, and noradrenergic neurotransmitter (NT) systems. RESULTS: The MKL model was able to significantly predict RD but not HA and NS from the brainstem metabolic activity. The MKL pattern regression model identified increased metabolic activity in the pontine nuclei and locus coeruleus, the medial reticular formation, the dorsal/median raphe, and the ventral tegmental area that contributed to the predictions of RD. CONCLUSIONS: The MKL algorithm identified a likely metabolic marker in the brainstem for RD in major depression. Although (18)FDG PET does not investigate specific NT systems, the predictive value of brainstem glucose metabolism on RD scores however indicates that this temperament dimension in the TRD state could be mediated by different monoaminergic systems, all involved in higher order reward-related behavior

Placebo aiTBS attenuates suicidal ideation and frontopolar cortical perfusion in major depression

The application of repetitive transcranial magnetic stimulation has been shown to rapidly decrease suicidal ideation in major depressive disorder (MDD). However, the neural working mechanisms behind this prompt attenuation of suicidal thoughts remains to be determined. Here, we examined how placebo-accelerated intermittent theta burst stimulation (aiTBS) may influence brain perfusion and suicidal thoughts using arterial spin labeling (ASL). In a randomized double-blind sham-controlled crossover trial, 45 MDD patients received aiTBS applied to the left dorsolateral prefrontal cortex (Trial registration: http://clinicaltrials.gov/show/NCT01832805). With each ASL scan measurement, suicidal ideation was assessed with the Beck Scale for Suicidal Ideation (BSI) and depression severity with the Beck Depression Inventory (BDI). Compared with active stimulation, the attenuation of suicidal ideation after 4 days of placebo aiTBS was related to significant frontopolar prefrontal perfusion decreases. These findings were unrelated to changes in depression severity scores. Although both active and sham aiTBS resulted in prompt decreases in suicidal ideation, specifically sham aiTBS significantly attenuated frontopolar perfusion in relation to reductions in BSI scores. Our findings show that in accelerated neurostimulation paradigms, placebo responses are related to perfusion decreases in brain areas associated with higher cognitive processes, resulting in suicidal ideation attenuation