Statistical computation of Boltzmann entropy and estimation of the optimal probability density function from statistical sample

In this work, we investigate the statistical computation of the Boltzmann entropy of statistical samples. For this purpose, we use both histogram and kernel function to estimate the probability density function of statistical samples. We find that, due to coarse-graining, the entropy is a monotonic increasing function of the bin width for histogram or bandwidth for kernel estimation, which seems to be difficult to select an optimal bin width/bandwidth for computing the entropy. Fortunately, we notice that there exists a minimum of the first derivative of entropy for both histogram and kernel estimation, and this minimum point of the first derivative asymptotically points to the optimal bin width or bandwidth. We have verified these findings by large amounts of numerical experiments. Hence, we suggest that the minimum of the first derivative of entropy be used as a selector for the optimal bin width or bandwidth of density estimation. Moreover, the optimal bandwidth selected by the minimum of the first derivative of entropy is purely data-based, independent of the unknown underlying probability density distribution, which is obviously superior to the existing estimators. Our results are not restricted to one-dimensional, but can also be extended to multivariate cases. It should be emphasized, however, that we do not provide a robust mathematical proof of these findings, and we leave these issues with those who are interested in them.Comment: 8 pages, 6 figures, MNRAS, in the pres

Effects of zero and reversed magnetic shear on resistive wall modes in a limiter tokamak plasma

Advanced tokamak scenarios often feature equilibriums with zero and reversed magnetic shear. To isolate and investigate their impacts on the resistive wall mode (RWM) instability analytically, we construct a series of cylindrical limiter equilibriums with reversed magnetic shear in the core and zero magnetic shear towards plasma edge, as a prototype of the configurations in advanced tokamak scenarios. Uniform plasma pressure is assumed, so that we can focus our analysis on the current-driven RWMs. Based on the reduced ideal MHD equations, analytical solutions for the $n=1$ resistive wall mode are obtained, which indicate that increasing the reversal of magnetic shear in the core region enhances the RWM instability, whereas the widened region of zero shear near edge leads to lower growth rate of RWM, except when the $q$ value with zero magnetic shear approaches rational values. On the other hand, enhanced positive shear at plasma edge is found to be stabilizing on RWM. NIMROD calculation results confirm these analytical findings

Exploration of the trajectory of being a breast cancer patient through collaborative writing in imaginal dialogue

This study aims to examine my personal lived experience, as a breast cancer patient, in navigating the liminality in the trajectory of illness. It focuses on developing understanding of the complexity in the thoughts, feelings and behaviours related to the (traumatic) cancer- related events that I have encountered over the past three years since diagnosis. The methodology is collaborative writing through imaginal dialogue (CW-ID) with Arthur Frank, the author of the books “At the Will of the Body” (2002 [1991]) and “The Wounded Storyteller” (2013 [1995]). It opens a creative and safe space for me to make sense of the life-threatening process. I discover that CW-ID, as a research methodology, supports me, as a researcher and cancer patient, in researching how the existential vulnerability of human beings is encountered and negotiated in the trajectory of illness. Surprisingly, CW-ID also manifests itself as therapeutic by making it possible for me to bring the overwhelming emotions in navigating the liminal spaces in the trajectory of illness to my supervisor. He bears witness to this creative-relational relationship which evolved over time between me, Frank, and others in both imaginal and real encounters in my trajectory of illness. The contribution of this study is that it offers an alternative to an autoethnographic approach in researching the deep, personal experience of health conditions that are biographically disruptive. It also opens an invitation to others who are interested in expanding the boundary of working collaboratively through imaginal dialogue with me and others. This can contribute to the knowledge creation of how human beings with medical conditions can develop a renewed sense of self in the process of navigating the liminal spaces in the trajectory of illness

Chandra observation of Abell 1689: New determination of mass distribution and comparison to lensing measurements

We present a new estimate of the projected X-ray mass of Abell 1689 observed with Chandra, in an attempt to clarify the issue of whether or not there exists a discrepancy between X-ray and gravitational lensing mass estimates claimed in previous investigations based on {\it Einstein}, {\it ROSAT} and {\it ASCA} observations. A particular attention is paid to examining if there is an offset between X-ray centroid and central dominant cD galaxy, which may be an indicator of the presence of local dynamical activities of intracluster gas in the central core and therefore, explain the discrepancy between X-ray and lensing mass estimates, if any. The unprecedentedly high spatial resolution achieved by Chandra allows us to precisely localize the X-ray centroid of Abell 1689, which appears to coincide perfectly with the central cD galaxy. This fact, along with the symmetry and regularity of the X-ray surface brightness and temperature distributions, suggests that Abell 1689 is a fully-relaxed cluster. We thus employ hydrostatic equilibrium hypothesis to determine the projected mass profile of Abell 1689, and compare it with the results obtained by different lensing techniques available in the literature. Our analysis confirms the existance of the discrepancy of a factor of $\sim2$ between X-ray and lensing mass estimates in the central region of $r\approx0.2$ Mpc, although the two methods yield essentially consistent result on large radii. If the perfect coincidence between the X-ray center and the cD galaxy of Abell 1689 detected by Chandra observation is not a projection effect, the central mass discrepancy between X-ray and lensing measurements may pose a challenge to our conventional understanding of dynamical evolution of the intracluster gas in the central regions of clusters.Comment: 26 pages, 7 figures, Accepted for publication in the Astrophysical Journa

Design of Self-Balancing Tracing Bicycle for Smart Car Competition Case Under Engineering Education

Smart car is an academic competition held for cultivating college students\u27 engineering ability in China for 16 years. To improve the performance of smart cars, this study integrates engineering education topics by introducing a smart car system with regard to the selection of key components, design of hardware and circuit boards, processing of sensor signals, as well as assembly, algorithms, and control. After completing this engineering education, students could achieve better results in the academic competition. According to the K model rules of the 16th smart car competition, a self-balancing autonomous tracking bicycle based on steering gear control is designed and developed. A gyroscope is used to detect the posture of the bicycle. It inductively receives the centerline of the track and then combined with the PID control algorithm realizes the autonomous tracking. The whole process from mechanical structure optimization and electronic circuit design to algorithm design, debugging, and competition runs through the CDIO of engineering education, realizing the cultivation of compound engineering innovative abilities

{μ-6,6′-Dimeth­oxy-2,2′-[ethane-1,2-diylbis(nitrilo­methyl­idyne)]diphenolato}-μ-nitrato-dinitratoholmium(III)zinc(II)

In the title heteronuclear ZnII–HoIII complex (systematic name: {μ-6,6′-dimeth­oxy-2,2′-[ethane-1,2-diylbis(nitrilo­methyl­idyne)]diphenolato-1κ4 O 1,O 1′,O 6,O 6′:2κ4 O 1,N,N′,O 1′)-μ-nitrato-1:2κ2 O:O′-dinitrato-1κ4 O,O′-holmium(III)zinc(II)), [HoZn(C18H18N2O4)(NO3)3], with the hexa­dentate Schiff base compartmental ligand N,N′-bis­(3-methoxy­salicyl­idene)ethyl­enediamine (H2 L), the Ho and Zn atoms are triply bridged by two phenolate O atoms of the Schiff base ligand and one nitrate ion. The five-coordinate Zn atom is in a square-pyramidal geometry with the donor centers of two imine N atoms, two phenolate O atoms and one of the bridging nitrate O atoms. The HoIII center has a ninefold coordination environment of O atoms, involving the phenolate O atoms, two meth­oxy O atoms, two O atoms from two nitrate ions and one from the bridging nitrate ion. Weak inter­molecular C—H⋯O inter­actions generate a two-dimensional double-layer structure