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We provide evidence that level repulsion in semiclassical spectrum is not just a feature of classically chaotic systems, but classically integrable systems as well. While in chaotic systems level repulsion develops on a scale of the mean level spacing, regardless of the location in the spectrum, in integrable systems it develops on a much longer scale — such as geometric mean of the mean level spacing and the running energy in the spectrum for hard wall billiards. We show that at this scale level correlations in integrable systems have a universal dependence on the level separation, as well as discuss their exact form at any scale. These correlations have dramatic consequences, including deviations from the Poissonian statistics in the nearest level spacing distribution and persistent oscillations of the level number variance over an energy interval as a function of the interval width. We illustrate our findings on two specific models — rectangular infinite well and a modified Kepler problem — that serve as generic types of a hard wall billiard and a potential problem without extra symmetries. Our theory and numerical work are based on the concept of parametric averaging that allows sampling of a statistical ensemble of integrable systems at a given spectral location (running energy).

We analyze two theoretical approaches to ensemble averaging for integrable systems in quantum chaos, spectral averaging (SA) and parametric averaging (PA). For SA, we introduce a new procedure, namely, rescaled spectral averaging (RSA). Unlike traditional SA, it can describe the correlation function of spectral staircase (CFSS) and produce persistent oscillations of the interval level number variance (IV). PA while not as accurate as RSA for the CFSS and IV, can also produce persistent oscillations of the global level number variance (GV) and better describes saturation level rigidity as a function of the running energy. Overall, it is the most reliable method for a wide range of statistics.

Some magnetic characteristics of Nd-Fe-B sintered magnets have been clarified, especially in the near vicinity within three times their length for cylindrical magnets. The flux densities as a function of distance (z) along the z-axis from the center of a single magnet were determined by utilizing Hall sensors. The repulsions between two magnets possessing identical shape were directly measured, also as a function of z, by using a home-made apparatus adopting piezoelectric device. The respective result has turned out well, coinciding with that of corresponding finite element method analysis and some analytical solutions. Also in this investigation, the correct formula directly applicable to the near H-field strength along the magnetic moment of a cylindrical magnet has been determined to be the exact solution that defies all confusing approximations or assumptions in theory, which were seen in textbooks or published in papers. Furthermore an analytical solution for the repulsion of magnet twins (cylindrical or hexagonal in shape), which can be handy in designing a variety of superconducting and/or electromagnetic devices, has been derived basically from Biot–Savart Law.

We discuss a first-order Cucker–Smale-type consensus model with attractive and repulsive interactions and present upper and lower bound estimates on the number of asymptotic point-clusters depending on the relative ranges of interactions and coupling strength. When the number of agents approaches infinity, we introduce a scalar conservation law with a non-local flux for a macroscopic description. We show that the corresponding conservation law admits a classical solution for sufficiently smooth initial data, which illustrates the shock avoidance effect due to the non-locality of the interactions. We also study the dynamics of special Dirac-Comb-type solutions consisting of two and three point-clusters.

There is a public and scholarly debate about whether personalized services of social-media platforms contribute to the rise of bipolarization of political opinions. On the one hand, it is argued that personalized services of online social networks generate filter bubbles limiting contact between users who disagree. This reduces opportunities for assimilative social influence between users from different camps and prevents opinion convergence. On the other hand, empirical research also indicated that exposing users to content from the opposite political spectrum can activate the counter-part of assimilative influence, repulsive influence. Fostering contact that leads to opinion assimilation and limiting contacts likely to induce repulsive interactions, it has been concluded, may therefore prevent bipolarization. With an agent-based model, we demonstrate here that these conclusions fail to capture the complexity that assimilative and repulsive influence generate in social networks. Sometimes, more assimilative influence can actually lead to more and not less opinion bipolarization. Likewise, increasing the exposure of users to like-minded individuals sometimes intensifies opinion polarization. While emerging only in specific parts of the parameter space, these counter-intuitive dynamics are robust, as our simulation experiments demonstrate. We discuss implications for the debate about filter bubbles and approaches to improve the design of online social networks. While we applaud the growing empirical research on the micro-processes of assimilative and repulsive influence in online settings, we warn that drawing conclusions about resulting macro-outcomes like opinion bipolarization requires a rigorous analysis capturing the complexity of online communication systems. Intuition alone is error-prone in this context. Accordingly, models capturing the complexity of social influence in networks should play a more important role in the design of communication systems.

Casimir energies and forces have been calculated in various configurations and boundary conditions. The calculations indicated that the Casimir energy might change its sign depending not only on the boundary conditions but also on geometry and topology of the configuration. With the development of nanotechnology, it is known that repulsive Casimir force is very important in nanodevices. In this paper, we review some research results on repulsive Casimir force, and discuss whether it could be realizable theoretically and experimentally.