We consider the Glauber dynamics of a ferromagnetic Ising-Kac model on a three-dimensional periodic lattice of size
Solomon rings, upholding the symbol of wisdom with profound historical roots, were widely used as decorations in ancient architecture and clothing. However, it was only recently discovered that such topological structures can be formed by self-organization in biological/chemical molecules, liquid crystals, etc. Here, we report the observation of polar Solomon rings in a ferroelectric nanocrystal, which consist of two intertwined vortices and are mathematically equivalent to a
- Award ID(s):
- 2133373
- PAR ID:
- 10538814
- Author(s) / Creator(s):
- ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; ; more »
- Publisher / Repository:
- Nature Portfolio
- Date Published:
- Journal Name:
- Nature Communications
- Volume:
- 14
- Issue:
- 1
- ISSN:
- 2041-1723
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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Abstract , in which the flipping rate of each spin depends on an average field in a large neighborhood of radius$$(2 N + 1)^3$$ . We study the random fluctuations of a suitably rescaled coarse-grained spin field as$$\gamma ^{-1}<\!\!< N$$ and$$N \rightarrow \infty $$ ; we show that near the mean-field value of the critical temperature, the process converges in distribution to the solution of the dynamical$$\gamma \rightarrow 0$$ model on a torus. Our result settles a conjecture from Giacomin et al. (1999). The dynamical$$\Phi ^4_3$$ model is given by a non-linear stochastic partial differential equation (SPDE) which is driven by an additive space-time white noise and which requires renormalisation of the non-linearity. A rigorous notion of solution for this SPDE and its renormalisation is provided by the framework of regularity structures (Hairer in Invent Math 198(2):269–504, 2014.$$\Phi ^4_3$$ https://doi.org/10.1007/s00222-014-0505-4 ). As in the two-dimensional case (Mourrat and Weber in Commun Pure Appl Math 70(4):717–812, 2017), the renormalisation corresponds to a small shift of the inverse temperature of the discrete system away from its mean-field value. -
Abstract The electric
E 1 and magneticM 1 dipole responses of the nucleus$$N=Z$$ Mg were investigated in an inelastic photon scattering experiment. The 13.0 MeV electrons, which were used to produce the unpolarised bremsstrahlung in the entrance channel of the$$^{24}$$ Mg($$^{24}$$ ) reaction, were delivered by the ELBE accelerator of the Helmholtz-Zentrum Dresden-Rossendorf. The collimated bremsstrahlung photons excited one$$\gamma ,\gamma ^{\prime }$$ , four$$J^{\pi }=1^-$$ , and six$$J^{\pi }=1^+$$ states in$$J^{\pi }=2^+$$ Mg. De-excitation$$^{24}$$ rays were detected using the four high-purity germanium detectors of the$$\gamma $$ ELBE setup, which is dedicated to nuclear resonance fluorescence experiments. In the energy region up to 13.0 MeV a total$$\gamma $$ is observed, but this$$B(M1)\uparrow = 2.7(3)~\mu _N^2$$ nucleus exhibits only marginal$$N=Z$$ E 1 strength of less than e$$\sum B(E1)\uparrow \le 0.61 \times 10^{-3}$$ fm$$^2 \, $$ . The$$^2$$ branching ratios in combination with the expected results from the Alaga rules demonstrate that$$B(\varPi 1, 1^{\pi }_i \rightarrow 2^+_1)/B(\varPi 1, 1^{\pi }_i \rightarrow 0^+_{gs})$$ K is a good approximative quantum number for Mg. The use of the known$$^{24}$$ strength and the measured$$\rho ^2(E0, 0^+_2 \rightarrow 0^+_{gs})$$ branching ratio of the 10.712 MeV$$B(M1, 1^+ \rightarrow 0^+_2)/B(M1, 1^+ \rightarrow 0^+_{gs})$$ level allows, in a two-state mixing model, an extraction of the difference$$1^+$$ between the prolate ground-state structure and shape-coexisting superdeformed structure built upon the 6432-keV$$\varDelta \beta _2^2$$ level.$$0^+_2$$ -
Abstract Consider two half-spaces
and$$H_1^+$$ in$$H_2^+$$ whose bounding hyperplanes$${\mathbb {R}}^{d+1}$$ and$$H_1$$ are orthogonal and pass through the origin. The intersection$$H_2$$ is a spherical convex subset of the$${\mathbb {S}}_{2,+}^d:={\mathbb {S}}^d\cap H_1^+\cap H_2^+$$ d -dimensional unit sphere , which contains a great subsphere of dimension$${\mathbb {S}}^d$$ and is called a spherical wedge. Choose$$d-2$$ n independent random points uniformly at random on and consider the expected facet number of the spherical convex hull of these points. It is shown that, up to terms of lower order, this expectation grows like a constant multiple of$${\mathbb {S}}_{2,+}^d$$ . A similar behaviour is obtained for the expected facet number of a homogeneous Poisson point process on$$\log n$$ . The result is compared to the corresponding behaviour of classical Euclidean random polytopes and of spherical random polytopes on a half-sphere.$${\mathbb {S}}_{2,+}^d$$ -
Abstract A well-known open problem of Meir and Moser asks if the squares of sidelength 1/
n for can be packed perfectly into a rectangle of area$$n\ge 2$$ . In this paper we show that for any$$\sum _{n=2}^\infty n^{-2}=\pi ^2/6-1$$ , and any$$1/2 that is sufficiently large depending on$$n_0$$ t , the squares of sidelength for$$n^{-t}$$ can be packed perfectly into a square of area$$n\ge n_0$$ . This was previously known (if one packs a rectangle instead of a square) for$$\sum _{n=n_0}^\infty n^{-2t}$$ (in which case one can take$$1/2 ).$$n_0=1$$ -
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