There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

抗原变异可使得多种致病微生物易于逃避宿主免疫应答。表达在感染红细胞表面的恶性疟原虫红细胞表面蛋白1（PfPMP1）与感染红细胞、内皮细胞、树突状细胞以及胎盘的单个或多个受体作用，在黏附及免疫逃避中起关键的作用。每个单倍体基因组var基因家族编码约60种成员，通过启动转录不同的var基因变异体为抗原变异提供了分子基础。

疟原虫var基因转换速率变化导致抗原变异[英]／Paul H, Robert P, Christodoulou Z, et al//Proc Natl Acad Sci U S A

Probability Distributions.- Linear Models for Regression.- Linear Models for Classification.- Neural Networks.- Kernel Methods.- Sparse Kernel Machines.- Graphical Models.- Mixture Models and EM.- Approximate Inference.- Sampling Methods.- Continuous Latent Variables.- Sequential Data.- Combining Models.

Pattern Recognition and Machine Learning

N G van Kampen 1981 Amsterdam: North-Holland xiv + 419 pp price Dfl 180 This is a book which, at a lower price, could be expected to become an essential part of the library of every physical scientist concerned with problems involving fluctuations and stochastic processes, as well as those who just enjoy a beautifully written book. It provides an extensive graduate-level introduction which is clear, cautious, interesting and readable.

https://iopscience.iop.org/article/10.1088/0031-9112/34/4/040/pdf

Stochastic Processes in Physics and Chemistry

Weyl and Dirac semimetals are three-dimensional phases of matter with gapless electronic excitations that are protected by topology and symmetry. As three-dimensional analogs of graphene, they have generated much recent interest. Deep connections exist with particle physics models of relativistic chiral fermions, and, despite their gaplessness, to solid-state topological and Chern insulators. Their characteristic electronic properties lead to protected surface states and novel responses to applied electric and magnetic fields. The theoretical foundations of these phases, their proposed realizations in solid-state systems, and recent experiments on candidate materials as well as their relation to other states of matter are reviewed.

Weyl and Dirac semimetals in three-dimensional solids

Liquid crystals are complex fluids that allow exquisite control of light propagation thanks to their orientational order and optical anisotropy. Inspired by recent advances in liquid-crystal photo-patterning technology, we propose a soft-matter platform for assembling topological photonic materials that holds promise for protected unidirectional waveguides, sensors, and lasers. Crucial to our approach is to use spatial variations in the orientation of the nematic liquid-crystal molecules to emulate the time modulations needed in a so-called Floquet topological insulator. The varying orientation of the nematic director introduces a geometric phase that rotates the local optical axes. In conjunction with suitably designed structural properties, this geometric phase leads to the creation of topologically protected states of light. We propose and analyze in detail soft photonic realizations of two iconic topological systems: a Su-Schrieffer-Heeger chain and a Chern insulator. The use of soft building blocks potentially allows for reconfigurable systems that exploit the interplay between topological states of light and the underlying responsive medium.

https://www.pnas.org/content/pnas/118/4/e2020525118.full.pdf

Liquid-crystal-based topological photonics

We present a theoretical study of the energetics of thin nematic shells with two charge-one-half defects and one charge-one defect. We determine the optimal arrangement: the defects are located on a great circle at the vertices of an isosceles triangle with angles of 66^{∘} at the charge-one-half defects and a distinct angle of 48^{∘}, consistent with experimental findings. We also analyze thermal fluctuations around this ground state and estimate the energy as a function of thickness. We find that the energy of the three-defect shell is close to the energy of other known configurations having two charge-one and four charge-one-half defects. This finding, together with the large energy barriers separating one configuration from the others, explains their observation in experiments as well as their long-time stability.

/pdf/spherical-nematic-shells-with-a-threefold-valence-6t6oixutrp.pdf

Spherical nematic shells with a threefold valence.

Statistical mechanics provides the foundation for describing complex materials using only a few thermodynamic variables. No such framework currently exists far from equilibrium. In this Letter, we demonstrate how thermodynamics emerges far from equilibrium, using fluids composed of active spinners as a case study. Activity gives rise to a single effective temperature that parameterizes both the equation of state and the emergent Boltzmann statistics. The same effective temperature, renormalized by velocity correlations, controls the linear response through canonical Green-Kubo relations for both the familiar shear viscosity and the odd (or Hall) viscosity observed in chiral fluids. The full frequency dependence of these viscosities can be derived analytically by modeling the active-spinner fluid as a random walker undergoing cyclotron motion in shear-stress space. More generally, we provide a first-principles derivation of the Green-Kubo relations valid for a broader class of fluids far from equilibrium. Besides advancing non-equilibrium thermodynamics, our work demonstrates in silico a non-invasive microrheology of active fluids.

Statistical mechanics of a chiral active fluid

Microscopic symmetries impose strong constraints on the elasticity of a crystalline solid. In addition to the usual spatial symmetries captured by the tensorial character of the elastic tensor, hidden nonspatial symmetries can occur microscopically in special classes of mechanical structures. Examples of such nonspatial symmetries occur in families of mechanical metamaterials where a duality transformation relates pairs of different configurations. We show on general grounds how the existence of nonspatial symmetries further constrains the elastic tensor, reducing the number of independent moduli. In systems exhibiting a duality transformation, the resulting constraints on the number of moduli are particularly stringent at the self-dual point but persist even away from it, in a way reminiscent of critical phenomena.

Symmetries and Dualities in the Theory of Elasticity.

Recent experiments have shown that colloidal crystals confined to weakly curved capillary bridges introduce groups of dislocations organized into ‘pleats’ as means to relieve the stress caused by the Gaussian curvature of the surface. We consider the onset of this curvature-screening mechanism, by examining the energetics of isolated dislocations and interstitials on capillary bridges with free boundaries. The boundary provides an essential contribution to the problem, akin to a background charge that “neutralizes” the unbalanced integrated curvature of the surface. This makes it favorable for topologically neutral dislocations and groups of dislocations – rather than topologically charged disclinations and scars – to relieve the stress caused by the unbalanced Gaussian curvature of the surface. This effect applies to any crystal on a surface with non-vanishing integrated Gaussian curvature and stress-free boundary conditions. We corroborate the analytic results by numerically computing the energetics of a defective lattice of springs confined to surfaces with weak positive and negative curvatures.

Vincenzo Vitelli

Papers

Liquid-crystal-based topological photonics

Spherical nematic shells with a threefold valence.

Statistical mechanics of a chiral active fluid

Symmetries and Dualities in the Theory of Elasticity.

Geometric background charge: dislocations on capillary bridges