Arithmetic of the moduli of hyperelliptic curves and principally polarized Abelian surfaces over global fields

We use geometric methods to establish an upper bound for counting stable hyperelliptic curves with a marked Weierstrass section ordered by height of discriminant at most $\mathcal{B}$ over $\mathbb{P}^{1}_{\mathbb{F}_q}$ with characteristic $p > 2g+1$; the acquired estimate is of order $\mathcal{O}_q\left( \mathcal{B}^{\frac{2g+3}{4g+2}} \right)$. We sharpen the estimate for each genus $g \ge 2$; specifically when $g=2$, this renders an estimate on the number of principally polarized Abelian surfaces over $\mathbb{F}_q(t)$. Through the global fields analogy, we formulate analogous new heuristics for counting stable hyperelliptic curves with a marked rational Weierstrass point or principally polarized Abelian surfaces over $\mathbb{Q}$. In Appendix, we determine the sharp estimate for counting elliptic curves with prescribed level structures or multiple marked points over $\mathbb{P}^{1}_{\mathbb{F}_q}$.Comment: 35 pages. The paper has been substantially reorganized throughout with a significantly improved Introduction. The Appendix (reinforced by Changho Han) has been integrated from arXiv:2002.06527. Comments welcome

Nature of self-diffusion in two-dimensional fluids

Self-diffusion in a two-dimensional simple fluid is investigated by both analytical and numerical means. We investigate the anomalous aspects of self-diffusion in two-dimensional fluids with regards to the mean square displacement, the time-dependent diffusion coefficient, and the velocity autocorrelation function using a consistency equation relating these quantities. We numerically confirm the consistency equation by extensive molecular dynamics simulations for finite systems, corroborate earlier results indicating that the kinematic viscosity approaches a finite, non-vanishing value in the thermodynamic limit, and establish the finite size behavior of the diffusion coefficient. We obtain the exact solution of the consistency equation in the thermodynamic limit and use this solution to determine the large time asymptotics of the mean square displacement, the diffusion coefficient, and the velocity autocorrelation function. An asymptotic decay law of the velocity autocorrelation function resembles the previously known self-consistent form, $1/(t\sqrt{\ln t})$, however with a rescaled time.Comment: 10 pages, to appear in New Journal of Physic

Thermodynamics of d-dimensional hard sphere fluids confined to micropores

We derive an analytical expression of the second virial coefficient of d-dimensional hard sphere fluids confined to slit pores by applying Speedy and Reiss’ interpretation of cavity space. We confirm that this coefficient is identical to the one obtained from the Mayer cluster expansion up to second order with respect to fugacity. The key step of both approaches is to evaluate either the surface area or the volume of the d-dimensional exclusion sphere confined to a slit pore. We, further, present an analytical form of thermodynamic functions such as entropy and pressure tensor as a function of the size of the slit pore. Molecular dynamics simulations are performed for d = 2 and d = 3, and the results are compared with analytically obtained equations of state. They agree satisfactorily in the low density regime, and, for given density, the agreement of the results becomes excellent as the width of the slit pore gets smaller, because the higher order virial coefficients become unimportant

High-speed and high-SNR photoacoustic microscopy based on a galvanometer mirror in non-conducting liquid

Optical-resolution photoacoustic microscopy (OR-PAM), a promising microscopic imaging technique with high ultrasound resolution and superior optical sensitivity, can provide anatomical, functional, and molecular information at scales ranging from the microvasculature to single red blood cells. In particular, real-time OR-PAM imaging with a high signal-to-noise ratio (SNR) is a prerequisite for widespread use in preclinical and clinical applications. Although several technical approaches have been pursued to simultaneously improve the imaging speed and SNR of OR-PAM, they are bulky, complex, not sensitive, and/or not actually real-time. In this paper, we demonstrate a simple and novel OR-PAM technique which is based on a typical galvanometer immersed in non-conducting liquid. Using an opto-ultrasound combiner, this OR-PAM system achieves a high SNR and fast imaging speed. It takes only 2 seconds to acquire a volumetric image with a wide field of view (FOV) of 4 x 8 mm(2) along the X and Y axes, respectively. The measured lateral and axial resolutions are 6.0 and 37.7 mu m, respectively. Finally, as a demonstration of the system's capability, we successfully imaged the microvasculature in a mouse ear in vivo. Our new method will contribute substantially to the popularization and commercialization of OR-PAM in various preclinical and clinical applications.11Ysciescopu

Accelerated identification of equilibrium structures of multicomponent inorganic crystals using machine learning potentials

The discovery of new multicomponent inorganic compounds can provide direct solutions to many scientific and engineering challenges, yet the vast size of the uncharted material space dwarfs current synthesis throughput. While the computational crystal structure prediction is expected to mitigate this frustration, the NP-hardness and steep costs of density functional theory (DFT) calculations prohibit material exploration at scale. Herein, we introduce SPINNER, a highly efficient and reliable structure-prediction framework based on exhaustive random searches and evolutionary algorithms, which is completely free from empiricism. Empowered by accurate neural network potentials, the program can navigate the configuration space faster than DFT by more than 10$^{2}$-fold. In blind tests on 60 ternary compositions diversely selected from the experimental database, SPINNER successfully identifies experimental (or theoretically more stable) phases for ~80% of materials within 5000 generations, entailing up to half a million structure evaluations for each composition. When benchmarked against previous data mining or DFT-based evolutionary predictions, SPINNER identifies more stable phases in the majority of cases. By developing a reliable and fast structure-prediction framework, this work opens the door to large-scale, unbounded computational exploration of undiscovered inorganic crystals.Comment: 3 figure

Molecular Hydrodynamics: Vortex Formation and Sound Wave Propagation

In the present study, quantitative feasibility tests of the hydrodynamic description of a two-dimensional fluid at the molecular level are performed, both with respect to length and time scales. Using high-resolution fluid velocity data obtained from extensive molecular dynamics simulations, we computed the transverse and longitudinal components of the velocity field by the Helmholtz decomposition and compared them with those obtained from the linearized Navier-Stokes (LNS) equations with time-dependent transport coefficients. By investigating the vortex dynamics and the sound wave propagation in terms of these field components, we confirm the validity of the LNS description for times comparable to or larger than several mean collision times. The LNS description still reproduces the transverse velocity field accurately at smaller times, but it fails to predict characteristic patterns of molecular origin visible in the longitudinal velocity field. Based on these observations, we validate the main assumptions of the mode-coupling approach. The assumption that the velocity autocorrelation function can be expressed in terms of the fluid velocity field and the tagged particle distribution is found to be remarkably accurate even for times comparable to or smaller than the mean collision time. This suggests that the hydrodynamic-mode description remains valid down to the molecular scale

Stimulated penetrating keratoplasty using real-time virtual intraoperative surgical optical coherence tomography

An intraoperative surgical microscope is an essential tool in a neuro-or ophthalmological surgical environment. Yet, it has an inherent limitation to classify subsurface information because it only provides the surface images. To compensate for and assist in this problem, combining the surgical microscope with optical coherence tomography (OCT) has been adapted. We developed a real-time virtual intraoperative surgical OCT (VISOCT) system by adapting a spectral-domain OCT scanner with a commercial surgical microscope. Thanks to our custommade beam splitting and image display subsystems, the OCT images and microscopic images are simultaneously visualized through an ocular lens or the eyepiece of the microscope. This improvement helps surgeons to focus on the operation without distraction to view OCT images on another separate display. Moreover, displaying the OCT live images on the eyepiece helps surgeon's depth perception during the surgeries. Finally, we successfully processed stimulated penetrating keratoplasty in live rabbits. We believe that these technical achievements are crucial to enhance the usability of the VISOCT system in a real surgical operating condition.open0