Regularity of H\"older continuous solutions of the supercritical quasi-geostrophic equation

We present a regularity result for weak solutions of the 2D quasi-geostrophic equation with supercritical ($\alpha< 1/2$) dissipation $(-\Delta)^\alpha$ : If a Leray-Hopf weak solution is H\"{o}lder continuous $\theta\in C^\delta({\mathbb R}^2)$ with $\delta>1-2\alpha$ on the time interval $[t_0, t]$, then it is actually a classical solution on $(t_0,t]$

BPS Wilson loops in Minkowski spacetime and Euclidean space

We give evidence that spacelike BPS Wilson loops do not exist in Minkowski spacetime. We show that spacelike Wilson loops in Minkowski spacetime cannot preserve any supersymmetries, in $d = 4$ $\mathcal N = 4$ super Yang-Mills theory, $d = 3$ $\mathcal N = 2$ super Chern-Simons-matter theory, and $d = 3$ $\mathcal N = 6$ Aharony-Bergman-Jafferis-Maldacena theory. We not only show this using infinite straight lines and circles as examples, but also we give proofs for general curves. We attribute this to the conflicts of reality conditions of the spinors. However, spacelike Wilson loops do exist in Euclidean space. There are both BPS Wilson loops along infinite straight lines and circular BPS Wilson loops. This is because the reality conditions of the spinors are lost after Wick rotation. The result is reasonable in view of the AdS/CFT correspondence.Comment: V1, 19 pages, no figure; V2, 25 pages, typos corrected and proofs for general curves added thanks to the careful reading and valuable suggestions of the anonymous referee, version to appear in EPJC; V3, 25 pages, published versio

Construction and classification of novel BPS Wilson loops in quiver Chern-Simons-matter theories

In this paper we construct and classify novel Drukker-Trancanelli (DT) type BPS Wilson loops along infinite straight lines and circles in $\mathcal N=2,3$ quiver superconformal Chern-Simons-matter theories, Aharony-Bergman-Jafferis-Maldacena (ABJM) theory, and $\mathcal N=4$ orbifold ABJM theory. Generally we have four classes of Wilson loops, and all of them preserve the same supersymmetries as the BPS Gaiotto-Yin (GY) type Wilson loops. There are several free complex parameters in the DT type BPS Wilson loops, and for two classes of Wilson loops in ABJM theory and $\mathcal N=4$ orbifold ABJM theory there are supersymmetry enhancements at special values of the parameters. We check that the differences of the DT type and GY type Wilson loops are $Q$-exact with $Q$ being some supercharges preserved by both the DT type and GY type Wilson loops. The results would be useful to calculate vacuum expectation values of the DT type Wilson loops in matrix models if they are still BPS quantum mechanically.Comment: V1, 28 pages; V2,35 pages, published versio

Exact results for Wilson loops in orbifold ABJM theory

We investigate the exact results for circular 1/4 and 1/2 BPS Wilson loops in the $d=3$ ${\mathcal N}=4$ super Chern-Simons-matter theory that could be obtained by orbifolding Aharony-Bergman-Jafferis-Maldacena (ABJM) theory. The partition function of the ${\mathcal N}=4$ orbifold ABJM theory has been computed previously in the literature. In this paper, we re-derive it using a slightly different method. We calculate the vacuum expectation values of the circular 1/4 BPS Wilson loops in fundamental representation and of circular 1/2 BPS Wilson loops in arbitrary representations. We use both the saddle point approach and Fermi gas approach. The results for Wilson loops are in accord with the available gravity results.Comment: V1, 19 pages, 1 figure; V2, 24 pages, 1 figure, general representations added; V3, 26 pages, no figure, published versio

Novel BPS Wilson loops in three-dimensional quiver Chern-Simons-matter theories

We show that generic three-dimensional $\mathcal N=2$ quiver super Chern-Simons-matter theories admit Bogomol'nyi-Prasad-Sommerfield (BPS) Drukker-Trancanelli (DT) type Wilson loops. We investigate both Wilson loops along timelike infinite straight lines in Minkowski spacetime and circular Wilson loops in Euclidean space. In Aharnoy-Bergman-Jafferis-Maldacena theory, we find that generic BPS DT type Wilson loops preserve the same number of supersymmetries as Gaiotto-Yin type Wilson loops. There are several free parameters for generic BPS DT type Wilson loops in the construction, and supersymmetry enhancement for Wilson loops happens for special values of the parameters.Comment: V1, 5 pages; V2, 13 pages, circular Wilson loops added, published versio

Analysis of consciousness for complete locked-in syndrome patients

This thesis presents methods for detecting consciousness in patients with complete locked-in syndrome (CLIS). CLIS patients are unable to speak and have lost all muscle movement. Externally, the internal brain activity of such patients cannot be easily perceived, but CLIS patients are considered to be still conscious and cognitively active. Detecting the current state of consciousness of CLIS patients is non-trivial, and it is difficult to ascertain whether CLIS patients are conscious or not. Thus, it is vital to develop alternative ways to re-establish communication with these patients during periods of awareness, and a possible platform is through brain–computer interface (BCI). Since consciousness is required to use BCI correctly, this study proposes a modus operandi to analyze not only in intracranial electrocorticography (ECoG) signals with greater signal-to-noise ratio (SNR) and higher signal amplitude, but also in non-invasive electroencephalography (EEG) signals. By applying three different time-domain analysis approaches sample entropy, permutation entropy, and Poincaré plot as feature extraction to prevent disease-related reductions of brainwave frequency bands in CLIS patients, and cross-validated to improve the probability of correctly detecting the conscious states of CLIS patients. Due to the lack a of 'ground truth' that could be used as teaching input to correct the outcomes, k-Means and DBSCAN these unsupervised learning methods were used to reveal the presence of different levels of consciousness for individual participation in the experiment first in locked-in state (LIS) patients with ALSFRS-R score of 0. The results of these different methods converge on the specific periods of consciousness of CLIS/LIS patients, coinciding with the period during which CLIS/LIS patients recorded communication with an experimenter. To determine methodological feasibility, the methods were also applied to patients with disorders of consciousness (DOC). The results indicate that the use of sample entropy might be helpful to detect awareness not only in CLIS/LIS patients but also in minimally conscious state (MCS)/unresponsive wakefulness syndrome (UWS) patients, and showed good resolution for both ECoG signals up to 24 hours a day and EEG signals focused on one or two hours at the time of the experiment. This thesis focus on consistent results across multiple channels to avoid compensatory effects of brain injury. Unlike most techniques designed to help clinicians diagnose and understand patients' long-term disease progression or distinguish between different disease types on the clinical scales of consciousness. The aim of this investigation is to develop a reliable brain-computer interface-based communication aid eventually to provide family members with a method for short-term communication with CLIS patients in daily life, and at the same time, this will keep patients' brains active to increase patients' willingness to live and improve their quality of life (QOL)