Geometric properties of the complete-graph Ising model in the loop representation

The exact solution of the Ising model on the complete graph (CG) provides an important, though mean-field, insight for the theory of continuous phase transitions. Besides the original spin, the Ising model can be formulated in the Fortuin-Kasteleyn random-cluster and the loop representation, in which many geometric quantities have no correspondence in the spin representations. Using a lifted-worm irreversible algorithm, we study the CG-Ising model in the loop representation, and, based on theoretical and numerical analyses, obtain a number of exact results including volume fractal dimensions and scaling forms. Moreover, by combining with the Loop-Cluster algorithm, we demonstrate how the loop representation can provide an intuitive understanding to the recently observed rich geometric phenomena in the random-cluster representation, including the emergence of two configuration sectors, two length scales and two scaling windows.Comment: 10 pages, 10 figure

Hydraulic loss analysis in a pump-turbine with special emphasis on local rigid vortex and shear

Since the pump as turbine is increasingly employed in energy storage, improving efficiency in both pump and turbine modes is required for economic benefits. This study aims to analyze vortex flow characteristics and vortex control methods in both modes to reduce hydraulic loss. In this paper, a delayed detached eddy simulation was applied in a low specific speed pump-turbine. Based on the entropy production analysis and vorticity binary decomposition in the local vortices, the results show that the local shear is the leading cause of hydraulic loss instead of the existence of vortices. The average wake loss can be 1.6 times higher than the loss in jet regions in pump mode, but there is little difference in the distribution of shear and vortices in the wake flow in turbine mode. The local loss caused by the rotor-stator interaction with a tongue effect at blade passing frequency is up to threefold over the loss without a tongue effect in both modes. Reducing shear and ratio of shear to rigid vorticity of the local vortices via modification in the volute tongue angle to suppress the tongue effect can be an effective way to decrease hydraulic loss in both modes

A study on the form of historical and cultural districts from the perspective of Conzenian School : as the Shengjin Town District in Nanchang City for an example

The protection of cultural relics and historic sites in my country began in the 1920s. In 1982, the State CounciI announced the first batch of 24 national historical and culturaI cities. In 1986, the State Council announced the second batch of historical and cultural cities to clearly protect historical districts (Gu,2003), according to my country The current policy standards divide the protection of historicaI and culturaI heritage into three IeveIs, namely: culturaI relics protection units, historical and cultural blocks, and historical and cultural cities (Wang, 2004 ). As a medium-scale cuIturaI heritage protection system, historicaI cultural blocks have gradually become a hot spot due to their relative culturaI integrity and operabiIity on scales, and multi-dimensionaI and comprehensive research has been launched (Dai et a I., 2012 ). In 2011, the People's Government of Jiangxi Province approved the three historicaI and culturaI districts including longevity PaIace (Wanshou PaIace), Shengjin Tower and jinxiancang Historic District in the protection plan for the famous historicaI and culturaI city of Nanchang. "The Nanchang Metropolitan Area Plan (2015-2030) was annouced in 2016" pointed out that the "One City, Three Streets, Four" historical and cultural blocks and historical features of Nanchang City should be protected. In 2017, the "Shengjin Tower Historical and CulturaI Block Protection Plan (2015-2020)" was proposed for the construction of the planned area. The land is changed to commerciaI land for unified commerciaI development. In the process of urban replacement, historical and cultural blocks are inevitably destroyed, leading to the loss of traditional style and existing historical structure. Conzenian School of Urban Morphology is used to study the historicaI and cultural blocks of Shengjin Tower and analyze its historical causes. The protection and re new a I have median level research significance

Finite-Size Scaling of the High-Dimensional Ising Model in the Loop Representation

Besides its original spin representation, the Ising model is known to have the Fortuin-Kasteleyn (FK) bond and loop representations, of which the former was recently shown to exhibit two upper critical dimensions $(d_c=4,d_p=6)$. Using a lifted worm algorithm, we determine the critical coupling as $K_c = 0.077\,708\,91(4)$ for $d=7$, which significantly improves over the previous results, and then study critical geometric properties of the loop-Ising clusters on tori for spatial dimensions $d=5$ to 7. We show that, as the spin representation, the loop Ising model has only one upper critical dimension at $d_c=4$. However, sophisticated finite-size scaling (FSS) behaviors, like two length scales, two configuration sectors and two scaling windows, still exist as the interplay effect of the Gaussian fixed point and complete-graph asymptotics. Moreover, using the Loop-Cluster algorithm, we provide an intuitive understanding of the emergence of the percolation-like upper critical dimension $d_p=6$ in the FK-Ising model. As a consequence, a unified physical picture is established for the FSS behaviors in all the three representations of the Ising model above $d_c=4$.Comment: 11 pages, 12 figure