Features of laser radiation interaction with metals having cryogenic temperature

In this paper we determine the features of the thermophysical processes involved in the interaction of laser radiation with metals that have cryogenic temperature. To do so, we use a one-dimensional model that involves heating a semi-infinite solid by a point thermal source with a constant flux density. Temperature fields, heating and cooling rates in the laser-irradiated zone for iron and titanium at the ambient temperatures of 77 (liquid nitrogen), 293 and 573 K were calculated. The intensity of the laser irradiation enabled the melting temperatures of 1933 K and 1812 K on the Ti and Fe surface, respectively, to be reached. The duration of the laser pulse was 4.5 ms. We show that a drop in ambient temperature from 573 to 77 K leads to a rise in cooling rate from 3.25 × 10³ and 6.4 × 10⁶ K/s to 4.25 × 10³ and 1.3 × 10⁷ K/s in the Ti and Fe targets, respectively. Agreement was good between the calculated depths of melting and phase transformation isotherms and the experimental depths of the interfaces of melting and heat-affected zones.     

Local wellposedness of nonlinear Maxwell equations with perfectly conducting boundary conditions

In this article we develop the local wellposedness theory for quasilinear Maxwell equations in $H^m$ for all $m\ge 3$ on domains with perfectly conducting boundary conditions. The macroscopic Maxwell equations with instantaneous material laws for the polarization and the magnetization lead to a quasilinear first order hyperbolic system whose wellposedness in $H^3$ is not covered by the available results in this case. We prove the existence and uniqueness of local solutions in $H^m$ with $m\ge 3$ of the corresponding initial boundary value problem if the material laws and the data are accordingly regular and compatible. We further characterize finite time blowup in terms of the Lipschitz norm and we show that the solutions depend continuously on their data. Finally, we establish the finite propagation speed of the solutions.     

NaK高位6~1∑~+态与H_2碰撞中转移能配置

脉冲激光激发NaK 2~1Σ~+←1~1Σ~+跃迁,单模Ti宝石激光器激发2~1Σ~+至高位态6~1Σ~+,研究了6~1Σ~+与H_2碰撞中的碰撞转移。3D→4P(1.7μm)和5S→4P(1.24μm)荧光发射说明了预解离和碰撞解离的产生。在不同的H_2密度下,通过以上能级的荧光测量得到了预解离率,碰撞解离及碰撞转移速率系数Γ_(3D)~P=(5.3±2.5)×10~8 s~(-1),Γ_(5S)~P=(3.1±1.5)×10~8 s~(-1),k_(3D)=(3.7±1.7)×10~(-11)cm~3·s~(-1),k_(5S)=(2.9±1.4)×10~(-11)cm~3·s~(-1),k_(4P→4S)=(1.1±0.5)×10~(-11)cm~3·s~(-1),k_(3D→4P)=(6.5±3.1)×10~(-12)cm~3·s~(-1),k_(5S→4P)=(4.1±1.9)×10~(-12)cm~3·s~(-1)在不同H_2密度下,记录时间分辨荧光,由Stern-Volmer公式得到6~1Σ~+→2~1Σ~+,2~1Σ~+→1~1Σ~+的自发辐射寿命分别为(28±10)ns和(15±4)ns。6~1Σ~+→2~1Σ~+6~1Σ~+→1~1Σ~+及2~1Σ~+→1~1Σ~+分子态间与H_2的碰撞转移速率系数分别为(1.8±0.6)×10~(-11)cm~3·s~(-1),(1.6±0.5)×10~(-10)cm~3·s~(-1)和(6.3±1.9)×10~(-11)cm~3·s~(-1)。转移到H_2的振动、转动和平动能各占总转移能的0.58,0.03和0.39。主要能量转移至振动和平动能,支持6~1Σ~+-H_2间的共线型碰撞机制。     

An analytic approximate method for solving stochastic integrodifferential equations

In this paper we compare the solution of a general stochastic integrodifferential equation of the Ito type, with the solutions of a sequence of appropriate equations of the same type, whose coefficients are Taylor series of the coefficients of the original equation. The approximate solutions are defined on a partition of the time-interval. The rate of the closeness between the original and approximate solutions is measured in the sense of the Lp-norm, so that it decreases if the degrees of these Taylor series increase, analogously to real analysis. The convergence with probability one is also proved.     

Diophantine approximation and badly approximable sets

Let (X,d) be a metric space and (Ω,d) a compact subspace of X which supports a non-atomic finite measure m. We consider ‘natural’ classes of badly approximable subsets of Ω. Loosely speaking, these consist of points in Ω which ‘stay clear’ of some given set of points in X. The classical set Bad of ‘badly approximable’ numbers in the theory of Diophantine approximation falls within our framework as do the sets Bad(i,j) of simultaneously badly approximable numbers. Under various natural conditions we prove that the badly approximable subsets of Ω have full Hausdorff dimension. Applications of our general framework include those from number theory (classical, complex, p-adic and formal power series) and dynamical systems (iterated function schemes, rational maps and Kleinian groups).     

Bifurcation of stable equilibria and nonlinear flux boundary condition with indefinite weight

We study bifurcation and stability of positive equilibria of a parabolic problem under a nonlinear Neumann boundary condition having a parameter and an indefinite weight. The main motivation is the selection migration problem involving two alleles and no gene flux acrossing the boundary, introduced by Fisher and Fleming, and Henry?s approach to the problem.Local and global structures of the set of equilibria are given. While the stability of constant equilibria is analyzed, the exponential stability of the unique bifurcating nonconstant equilibrium solution is established. Diagrams exhibiting the bifurcation and stability structures are also furnished. Moreover the asymptotic behavior of such solutions on the boundary of the domain, as the positive parameter goes to infinity, is also provided.The results are obtained via classical tools like the Implicit Function Theorem, bifurcation from a simple eigenvalue theorem and the exchange of stability principle, in a combination with variational and dynamical arguments.     

Classical and singular stochastic control for the optimal dividend policy when there is regime switching

Motivated by economic and empirical arguments, we consider a company whose cash surplus is affected by macroeconomic conditions. Specifically, we model the cash surplus as a Brownian motion with drift and volatility modulated by an observable continuous-time Markov chain that represents the regime of the economy. The objective of the management is to select the dividend policy that maximizes the expected total discounted dividend payments to be received by the shareholders. We study two different cases: bounded dividend rates and unbounded dividend rates. These cases generate, respectively, problems of classical stochastic control with regime switching and singular stochastic control with regime switching. We solve these problems, and obtain the first analytical solutions for the optimal dividend policy in the presence of business cycles. We prove that the optimal dividend policy depends strongly on macroeconomic conditions.     

Multi‐peak standing waves for nonlinear Schrödinger equations involving critical growth

For the following singularly perturbed problem we construct a solution which concentrates at several given isolated positive local minimum components of V as . Here, the nonlinearity f is of critical growth. Moreover, the monotonicity of and the so‐called Ambrosetti–Rabinowitz condition are not required.     

Unidirectional evolution equations of diffusion type

This paper is concerned with the uniqueness, existence, partial smoothing effect, comparison principle and long-time behavior of solutions to the initial-boundary value problem for a unidirectional diffusion equation. The unidirectional evolution often appears in Damage Mechanics due to the strong irreversibility of crack propagation or damage evolution. The existence of solutions is proved in an $L^2$-framework by employing a backward Euler scheme and by introducing a new method of a priori estimates based on a reduction of discretized equations to variational inequalities of obstacle type and by developing a regularity theory for such obstacle problems. The novel discretization argument will be also applied to prove the comparison principle as well as to investigate the long-time behavior of solutions.     

On the existence of weak solutions of semilinear elliptic equations and systems with Hardy potentials

Let $\mathrm{\Omega }?{\mathbb{R}}^N$ ($N\ge 3$) be a bounded $C^2$ domain and $\delta (x)=\mathrm{dist}(x,?\mathrm{\Omega })$. Put $L_{\mu }=\mathrm{\Delta }+\frac{\mu }{{\delta }^2}$ with $\mu >0$. In this paper, we provide various necessary and sufficient conditions for the existence of weak solutions to

$?L_{\mu }u=u^p+\tau \text{in }\mathrm{\Omega },u=\nu \text{on }?\mathrm{\Omega },$

where $\mu >0$, $p>0$, τ and ν are measures on Ω and ?Ω respectively. We then establish existence results for the system

$\{\begin{array}{cc}\hfill & ?L_{\mu }u=?v^p+\tau \text{in }\mathrm{\Omega },\hfill \\ \hfill & ?L_{\mu }v=?u^{\stackrel{?}{p}}+\stackrel{?}{\tau }\text{in }\mathrm{\Omega },\hfill \\ \hfill & u=\nu ,v=\stackrel{?}{\nu }\text{on }?\mathrm{\Omega },\hfill \end{array}$

where $?=\pm 1$, $p>0$, $\stackrel{?}{p}>0$, τ and $\stackrel{?}{\tau }$ are measures on Ω, ν and $\stackrel{?}{\nu }$ are measures on ?Ω. We also deal with elliptic systems where the nonlinearities are more general.