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

https://math.berkeley.edu/~sethian/Papers/sethian.osher.88.pdf

Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations

This paper reviews recent experimental and theoretical progress concerning many-body phenomena in dilute, ultracold gases. It focuses on effects beyond standard weak-coupling descriptions, such as the Mott-Hubbard transition in optical lattices, strongly interacting gases in one and two dimensions, or lowest-Landau-level physics in quasi-two-dimensional gases in fast rotation. Strong correlations in fermionic gases are discussed in optical lattices or near-Feshbach resonances in the BCS-BEC crossover.

/pdf/many-body-physics-with-ultracold-gases-552s10zfdn.pdf

Many-Body Physics with Ultracold Gases

A comprehensive review of spatiotemporal pattern formation in systems driven away from equilibrium is presented, with emphasis on comparisons between theory and quantitative experiments. Examples include patterns in hydrodynamic systems such as thermal convection in pure fluids and binary mixtures, Taylor-Couette flow, parametric-wave instabilities, as well as patterns in solidification fronts, nonlinear optics, oscillatory chemical reactions and excitable biological media. The theoretical starting point is usually a set of deterministic equations of motion, typically in the form of nonlinear partial differential equations. These are sometimes supplemented by stochastic terms representing thermal or instrumental noise, but for macroscopic systems and carefully designed experiments the stochastic forces are often negligible. An aim of theory is to describe solutions of the deterministic equations that are likely to be reached starting from typical initial conditions and to persist at long times. A unified description is developed, based on the linear instabilities of a homogeneous state, which leads naturally to a classification of patterns in terms of the characteristic wave vector q0 and frequency ω0 of the instability. Type Is systems (ω0=0, q0≠0) are stationary in time and periodic in space; type IIIo systems (ω0≠0, q0=0) are periodic in time and uniform in space; and type Io systems (ω0≠0, q0≠0) are periodic in both space and time. Near a continuous (or supercritical) instability, the dynamics may be accurately described via "amplitude equations," whose form is universal for each type of instability. The specifics of each system enter only through the nonuniversal coefficients. Far from the instability threshold a different universal description known as the "phase equation" may be derived, but it is restricted to slow distortions of an ideal pattern. For many systems appropriate starting equations are either not known or too complicated to analyze conveniently. It is thus useful to introduce phenomenological order-parameter models, which lead to the correct amplitude equations near threshold, and which may be solved analytically or numerically in the nonlinear regime away from the instability. The above theoretical methods are useful in analyzing "real pattern effects" such as the influence of external boundaries, or the formation and dynamics of defects in ideal structures. An important element in nonequilibrium systems is the appearance of deterministic chaos. A greal deal is known about systems with a small number of degrees of freedom displaying "temporal chaos," where the structure of the phase space can be analyzed in detail. For spatially extended systems with many degrees of freedom, on the other hand, one is dealing with spatiotemporal chaos and appropriate methods of analysis need to be developed. In addition to the general features of nonequilibrium pattern formation discussed above, detailed reviews of theoretical and experimental work on many specific systems are presented. These include Rayleigh-Benard convection in a pure fluid, convection in binary-fluid mixtures, electrohydrodynamic convection in nematic liquid crystals, Taylor-Couette flow between rotating cylinders, parametric surface waves, patterns in certain open flow systems, oscillatory chemical reactions, static and dynamic patterns in biological media, crystallization fronts, and patterns in nonlinear optics. A concluding section summarizes what has and has not been accomplished, and attempts to assess the prospects for the future.

/pdf/pattern-formation-outside-of-equilibrium-1601b9znum.pdf

Pattern formation outside of equilibrium

Several common modes of crystal growth provide particularly simple and elegant examples of spontaneous pattern formation in nature. Phenomena of interest here are those in which an advancing nonfaceted solidification front suffers an instability and subsequently reorganizes itself into a more complex mode of behavior. The purpose of this essay is to examine several such situations and, in doing this, to identify a few new theoretical ideas and a larger number of outstanding problems. The systems studied are those in which solidification is controlled entirely by a single diffusion process, either the flow of latent heat away from a moving interface or the analogous redistribution of chemical constituents. Convective effects are ignored, as are most effects of crystalline anisotropy. The linear theory of the Mullins-Sekerka instability is reviewed for simple planar and spherical cases and also for a special model of directional solidification. These techniques are then extended to the case of a freely growing dendrite, and it is shown how this analysis leads to an understanding of sidebranching and tip-splitting instabilities. A marginal-stability hypothesis is introduced; and it is argued that this intrinsically nonlinear theory, if valid, permits aone to use results of linear-stability analysis to predict dendritic growth rates. The review concludes with a discussion of nonlinear effects in directional solidication. The nonplanar, cellular interfaces which emerge in this situation have much in common with convection patterns in hydrodynamics. The cellular stability problem is discussed briefly, and some preliminary attempts to do calculations in the strongly nonlinear regime are summarized.

Instabilities and pattern formation in crystal growth

We propose a dynamical theory of low-temperature shear deformation in amorphous solids. Our analysis is based on molecular-dynamics simulations of a two-dimensional, two-component noncrystalline system. These numerical simulations reveal behavior typical of metallic glasses and other viscoplastic materials, specifically, reversible elastic deformation at small applied stresses, irreversible plastic deformation at larger stresses, a stress threshold above which unbounded plastic flow occurs, and a strong dependence of the state of the system on the history of past deformations. Microscopic observations suggest that a dynamically complete description of the macroscopic state of this deforming body requires specifying, in addition to stress and strain, certain average features of a population of two-state shear transformation zones. Our introduction of these state variables into the constitutive equations for this system is an extension of earlier models of creep in metallic glasses. In the treatment presented here, we specialize to temperatures far below the glass transition and postulate that irreversible motions are governed by local entropic fluctuations in the volumes of the transformation zones. In most respects, our theory is in good quantitative agreement with the rich variety of phenomena seen in the simulations. {copyright} {ital 1998} {ital The American Physical Society}

https://arxiv.org/pdf/cond-mat/9712114.pdf

Dynamics of viscoplastic deformation in amorphous solids

By considering a model in which charge is transported via phonon-induced tunneling of electrons between localized states which are randomly distributed in energy and position, Mott has obtained an electrical conductivity of the form $\ensuremath{\sigma}\ensuremath{\propto}\mathrm{exp}[\ensuremath{-}{(\frac{\ensuremath{\lambda}{\ensuremath{\alpha}}^{3}}{{\ensuremath{\rho}}_{0}\mathrm{kT}})}^{\frac{1}{4}}]$. Here $T$ is the temperature of the system, ${\ensuremath{\rho}}_{0}$ is the density of states at the Fermi level, $\ensuremath{\lambda}$ is a dimensionless constant, and ${\ensuremath{\alpha}}^{\ensuremath{-}1}$ is the distance for exponential decay of the wave functions. We rederive these results, relating $\ensuremath{\lambda}$ to the critical density of a certain dimensionless percolation problem, and we estimate $\ensuremath{\lambda}$ to be approximately 16. The applicability of the model to experimental observations on amorphous Ge, Si, and C is discussed.

James S. Langer

Papers

Instabilities and pattern formation in crystal growth

Dynamics of viscoplastic deformation in amorphous solids

Hopping Conductivity in Disordered Systems

Statistical theory of the decay of metastable states

Theory of the condensation point