In the present paper we study a type of generic singularity of mean curvature flow modelled on the bubble-sheet S1xR3, 1 x R 3 , and we derive an asymptotic profile for a neighbourhood of singularity.

In this paper we study a neighborhood of generic singularities formed by mean curvature flow (MCF). We limit our consideration to the singularities modelled on $\mathbb{S}^3\times\mathbb{R}$ because, compared to the cases $\mathbb{S}^k\times \mathbb{R}^{l}$ with $l\geq 2$, the present case has the fewest possibilities to be considered. For various possibilities, we provide a detailed description for a small, but fixed, neighborhood of singularity, and prove that a small neighborhood of the singularity is mean convex, and the singularity is isolated. For the remaining possibilities, we conjecture that an entire neighborhood of the singularity becomes singular at the time of blowup, and present evidences to support this conjecture. A key technique is that, when looking for a dominating direction for the rescaled MCF, we need a normal form transformation, as a result, the rescaled MCF is parametrized over some chosen curved cylinder, instead over a standard straight one. This is a long paper. The introduction is carefully written to present the key steps and ideas.

Ideas and results in the quantum theory of experiments are reviewed. To fix ideas, a concrete example of indirect measurements, an experiment devised by Guerlin et al. (Nature 448(7156):889–893, 2007), and theoretical interpretations thereof (Bauer and Bernard, Phys Rev A 84(4):044103, 2011; Bauer et al., Ann H Poincaré 14:639–679, 2013) are recalled. Subsequently two important elements of the Copenhagen interpretation of quantum mechanics, viz. the von Neumann- and the Lüders measurement postulates, are recalled and rendered more precise. Next, a model originally proposed by Gisin et al. (Phys Rev Lett 52:1657–1660, 1984) is described and shown to imply these postulates. It is then used to provide a theoretical description of the experiment in Guerlin et al. (Nature 448(7156):889–893, 2007) involving a “Heisenberg cut” differing from the one invoked in (Bauer and Bernard, Phys Rev A 84(4):044103, 2011; Bauer et al., Ann H Poincaré 14:639–679, 2013; J Stat Phys 162:924–958, 2016). Some technical issues in the analysis of Gisin’s model are elaborated upon. The paper concludes with remarks on a general principle that implies a universal law governing the stochastic time evolution of states of individual physical systems featuring events and leading to a solution of the so-called measurement problem in quantum mechanics.

A proposal of how to complete non-relativistic quantum mechanics to a physically meaningful, mathematically precise and logically coherent theory is reviewed. Our proposal leads to a general, non-linear stochastic law for the time-evolution of states of individual physical systems. An application of the general formalism to the quantum theory of fluorescence of an atom coupled to the radiation field is sketched. Some remarks on relativistic quantum theory conclude our review.

We study formation of generic singularities under mean curvature flow by combining the different approaches and results, namely the techniques used by the author and the collaborators, where we obtained detailed information when the initial hypersurfaces are close to cylinders, and these results by Colding and Minicozzi, which included all the generic blowups. Here we choose to study the cases where the rescaled flow converges to the cylinder S1×R3, and we extend the region controlled by Colding and Minicozzi to find a finer description of a neighborhood of the singularity.

It is wellknown that the eigenvalues em bedded in a continuous spectrum of a self-adjoint operator, L, are unstable under generic perturbations. Intuitively,one can think oftim e periodic solutions (‘bound states’) ofthe equation,i_ u = Lu,corresponding to such eigenvalues,leaking their energy to the continuousspectrum solutions(‘scattering states’)which in turn isradiated to in nity.Thecouplingbetween tim e-periodicand continuousspectrum solutions wascom puted in the second orderofperturbation theory by P.A.M .Dirac in 1927 (see [Di],resulting in an elegantexpression.Ifthisexpression isnonzero, then the decay m entioned above takes place. E.Ferm icalled this condition \G olden Rule No. 2". In physics literature it is known as the Ferm iG olden Rule(FG R).TheFG R,com puted rstforatom icand m olecularstates,appears in m any otherareasofphysicse.g.in non-equilibrium statisticalm echanics.O f course,due to energy conservation the isolated eigenvalues are stable under reasonabletim e independentperturbations. TheFerm iG olden Rulewasintroduced tononlinearHam iltonian PDE in [S] whereitwasused toprovethattim e-periodicsolutionsforlinear(and nonlinear) waveequationsareunstableundergenericnonlinearperturbations.Itwasused

We discuss a one-dimensional version of the Landau–Pekar equations, which are a system of coupled differential equations with two different time scales. We derive an approximation on the slow time scale in the spirit of a non-linear adiabatic theorem. Dispersive estimates for solutions of the Schrödinger equation with time-dependent potential are a key technical ingredient in our proof.

We consider the rate of convergence of solutions of spatially inhomogeneous Boltzmann equations, with hard-sphere potentials, to some equilibriums, called Maxwellians. Maxwellians are spatially homogeneous static Maxwell velocity distributions with different temperatures and mean velocities. We study solutions in weighted space [Formula: see text]. The result is that, assuming the solution is sufficiently localized and sufficiently smooth, then the solution, in [Formula: see text]-space, converges to a Maxwellian, exponentially fast in time.

Symplectic methods, including the generating function method, the symplectic Runge-Kutta (RK) method, the symplectic partitioned Runge-Kutta method, the multi-step method and so on, are applicable to Hamiltonian systems. They can preserve the symplectic structure in the phase space and the laws of the Hamiltonian system. But in the time domain, due to phase lags in the computing course, the RK methods and the symplectic methods have the same algebraic precision under the same algebraic order of schemes. After longtime computing, the numerical precision goes worse and worse in the time domain. To improve the precision, a new method combining the highly precise direct integration method with the symplectic difference scheme, called the HPD-symplectic method, was proposed. This method, proved to be symplectic, can preserve the symplectic structure. Moreover, the HPD-symplectic method can largely decrease the phase error in the time domain, and accordingly, improve the numerical precision even up to an error level of 10-13. For systems with mixed frequencies or rigid systems, the traditional symplectic methods can hardly work well, while the HPD-symplectic method can simulate the signals at both high and low frequencies well with large time steps but no additional computation cost. The results of numerical examples demonstrate the reliability and effectiveness of the proposed method.

In the present paper we study a type of generic singularity of mean curvature flow modelled on the bubble-sheet 𝕊^1×ℝ^3 , and we derive an asymptotic profile for a neighborhood of singularity.

We consider one of the generic regimes of formation of singularities, and we obtain a detailed description of a possibly small, but fixed, neighborhood of the blowup point, up to (and including) the blowup time. Estimates up to any order of derivatives are provided. They are small up to a proper rescaling, if sufficiently close to the blowup time and the blowup point. And we find that the singularity is isolated from the other ones.

We consider one of the generic regimes of formation of singularities, and we obtain a detailed description of a possibly small, but fixed, neighborhood of the blowup point, up to (and including) the blowup time, and find that it is mean convex in the considered set. This confirms a conjecture by Ilmanen. Estimates up to any order of derivatives are provided. They are small up to a proper rescaling, if sufficiently close to the blowup time and the blowup point. And we find that the singularity is isolated from the other ones.

We study surfaces evolving by mean curvature flow (MCF). For an open set of initial data that are $C^3$-close to round, but without assuming rotational symmetry or positive mean curvature, we show that MCF solutions become singular in finite time by forming neckpinches, and we obtain detailed asymptotics of that singularity formation. Our results show in a precise way that MCF solutions become asymptotically rotationally symmetric near a neckpinch singularity.

Frohlich's polaron Hamiltonian describes an electron coupled to the quantized phonon field of an ionic crystal. We show that in the strong coupling limit the dynamics of the polaron are approximated by an effective nonlinear partial differential equation due to Landau and Pekar, in which the phonon field is treated as a classical field.

Dynamics response of systems to impact or loading may be effectively treated by direct integration. However, it is often difficult to select the time-step of integration properly, especially in the case which the system is badly stiff. High Precision Direct integration based on variational principle is given (HPD-VP) for homogeneous systems and HHPD-VP method for the nonhomogeneous systems are given. This method not only takes the advantage of variational principle formula, which is much precise an d is stiff A-stable, but also can avoid the truncation error of the computer. For the large systems, especially, the systems with different frequency or the stiff systems, our methods are stable, accurate and efficient. Numerical experiments show the convergence order of the scheme derived from the variational principle, and is much precise and is effective in engineering.

哈密顿系统是一类重要的动力系统,针对哈密顿系统,设计出多类辛方法:SRK、SPRK、辛多步法、生成函数法等.长久以来数值方法在求解哈密顿系统过程中辛特性和保能量特性不能得到同时满足,近年来提出的有限元方法,对于线性系统具有保辛和保能量的优良特性.但是,以上方法都存在相位漂移(轨道偏离)现象,长时间仿真,计算效果会大打折扣.提出精细辛有限元方法(HPD-FEM)求解哈密顿系统,该方法继承时间有限元方法求解哈密顿系统所具有的保哈密顿系统的辛结构和哈密顿函数守恒性的优良特性,同时,通过精细化时间步长极大地减小了时间有限元方法的相位误差.HPD-FEM相较与针对相位误差专门设计的计算格式FSJS、RKN以及SRPK方法具有更好的纠正效果,几乎达到机器精度,误差为O(10-13),同时,HPD-FEM克服了FSJS、RKN和SPRK方法不能保证哈密顿函数守恒的缺点.对于高低混频系统和刚性系统,常规算法很难在较大步长下,同时实现对高低频精确仿真,HPD-FEM通过精细计算时间步长,在大步长情况下,实现高低混频的精确仿真.HPD-FEM方法在计算过程中精细方法没有额外增加计算量,计算效率高.数值结果显示本文提出的方法切实有效.

对于线性Hamilton系统,辛差分方法可以保持系统的辛结构,有限元方法可以保证系统的辛性质并具有能量守恒特性.但辛差分方法和有限元方法时域上仍然存在相位误差,使得计算的精度不是很理想.提出极小化相位误差加权间断有限元辛方法(WDG-PF),该方法是辛方法,同时,对Hamilton系统的求解具有极小的相位误差.数值显示该方法可以保证Hamilton系统的能量守恒性.WDG-PF方法解决了时间有限元方法(TFE)存在的相位漂移现象,同时指出间断有限元方法可以通过加权处理达到保辛要求.WDG-PF方法相较于针对相位误差设计的计算格式分数步对称辛算法(FSJS)、辛Runge-Kutta-Nystrom(RKN)格式以及辛分块Runge-Kutta(SPRK)等方法,WDG-PF显著地减少相位误差,和显著提高Hamilton系统能量精度的优点.相位误差和能量误差几乎达到计算机精度.同时单元内部具有超收敛现象.特别针对高低混频Hamilton系统,传统方法很难在固定的步长下同时实现对高频和低频信号的精确仿真,WDG-PF方法则可以在大步长下同时实现对低频信号和高频信号的高精度仿真.数值显示,WDG-PF方法切实有效.

唐氏综合征(DownSyndrome,DS)产前筛查不仅涉及标志物生化检测过程,还涉及生物统计学数据处理过程。本文旨在介绍DS筛查生物统计学基本概念,以及基于生物统计学原则的质量控制和评价方法,帮助建立全程覆盖的DS筛查质量控制和评价体系,促进DS筛查质量进一步提高。

We study noncompact surfaces evolving by mean curvature flow. Without any symmetry assumptions, we prove that any solution that is C-3-close at some time to a standard neck will develop a neckpinch singularity in finite time, will become asymptotically rotationally symmetric in a space-time neighborhood of its singular set, and will have a unique tangent flow.

In this paper we consider a resonance problem, in a generic regime, in the consideration of relaxation of ground states of semilinear Schrodinger equations. Different from previous results, our consideration includes the presence of resonance, resulted by overlaps of frequencies of different states. All the known key results, proved under non-resonance conditions, have been recovered uniformly. These are achieved by better understandings of normal form transformation and Fermi Golden rule. Especially, we find that if certain denominators are zeros (or small), resulted by the presence of resonances (or close to it), then cancellations between terms make the corresponding numerators proportionally small.