Convergence of minimum norm elements of projections and intersections of nested affine spaces in Hilbert space

We consider a Hilbert space, an orthogonal projection onto a closed subspace and a sequence of downwardly directed affine spaces. We give sufficient conditions for the projection of the intersection of the affine spaces into the closed subspace to be equal to the intersection of their projections. Under a closure assumption, one such (necessary and) sufficient condition is that summation and intersection commute between the orthogonal complement of the closed subspace, and the subspaces corresponding to the affine spaces. Another sufficient condition is that the cosines of the angles between the orthogonal complement of the closed subspace, and the subspaces corresponding to the affine spaces, be bounded away from one. Our results are then applied to a general infinite horizon, positive semi-definite, linear quadratic mathematical programming problem. Specifically, under suitable conditions, we show that optimal solutions exist and, modulo those feasible solutions with zero objective value, they are limits of optimal solutions to finite-dimensional truncations of the original problem.     

Nonadiabatic quantum dynamics of Br2 in solid Ar: A four-dimensional study of the B to C state predissociation

Quantum dynamics simulations are performed for a diatomics-in-molecules based model of Br₂ in solid Ar which incorporates four nuclear degrees of freedom and four electronic states. The nuclear motions comprise two large amplitude coordinates describing the Br₂ bond distance and an effective symmetry-preserving matrix mode. Two symmetry-lowering harmonic modes are added in the spirit of linear vibronic coupling theory. Initiating the dynamics on the B state by means of an ultrafast laser pulse, nonadiabatic transitions to the two degenerate C states are monitored and the effect of vibrational preexcitation in the electronic ground state is investigated.     

Guaranteed consistency of surface intersections and trimmed surfaces using a coupled topology resolution and domain decomposition scheme

We describe a method that serves to simultaneously determine the topological configuration of the intersection curve of two parametric surfaces and generate compatible decompositions of their parameter domains, that are amenable to the application of existing perturbation schemes ensuring exact topological consistency of the trimmed surface representations. To illustrate this method, we begin with the simpler problem of topology resolution for a planar algebraic curve F(x,y)=0 in a given domain, and then extend concepts developed in this context to address the intersection of two tensor-product parametric surfaces p(s,t) and q(u,v) defined on (s,t)∈[0,1]² and (u,v)∈[0,1]². The algorithms assume the ability to compute, to any specified precision, the real solutions of systems of polynomial equations in at most four variables within rectangular domains, and proofs for the correctness of the algorithms under this assumption are given. Mathematics subject classification (2000)  65D17     

不重叠型Schwarz交替法的加速收敛

Matsokin与Nepomnyaschikh所提出的不重叠型S交替法的算法中不含有松弛因子，我们知道它有与h无关的几何收敛速度，但由于不含算法参数，该算法不能根据具体的情况加速收敛，本文提出加速收敛算法，我们在原算法的基础上引入两个松弛因子θ2，θ2并证明了除了例外均可实现加速收敛，θ1=θ^-1，θ2=θ^-2是满足均衡条件的最佳松弛因子，最后的算例表明该加速算法的有效性。     

Scattering theory and the Aharonov–Bohm effect in quasiclassical physics

Scattering of a nonrelativistic quantum-mechanical particle by an impenetrable magnetic vortex is considered. The nonvanishing transverse size of the vortex is taken into account, and the limit of short, as compared to this size, wavelengths of the scattered particle is analyzed. We show that the scattering Aharonov–Bohm effect persists in the quasiclassical limit owing to the diffraction persisting in the short-wavelength limit. As a result, the vortex flux serves as a gate for the propagation of short-wavelength, almost classical, particles. This quasiclassical effect is more feasible to experimental detection in the case when space outside the vortex is conical.     

Intersections of graphs

The study of graph homomorphisms has a long and distinguished history, with applications in many areas of graph theory. There has been recent interest in counting homomorphisms, and in particular on the question of finding upper bounds for the number of homomorphisms from a graph G into a fixed image graph H. We introduce our techniques by proving that the lex graph has the largest number of homomorphisms into K₂ with one looped vertex (or equivalently, the largest number of independent sets) among graphs with fixed number of vertices and edges. Our main result is the solution to the extremal problem for the number of homomorphisms into P, the completely looped path of length 2 (known as the Widom–Rowlinson model in statistical physics). We show that there are extremal graphs that are threshold, give explicitly a list of five threshold graphs from which any threshold extremal graph must come, and show that each of these “potentially extremal” threshold graphs is in fact extremal for some number of edges. Copyright © 2011 Wiley Periodicals, Inc. J Graph Theory 67: 261–284, 2011     

Two-photon polymerisation of novel shapes using a conically diffracted femtosecond laser beam

Sub-micron ring, pillar and wall structures were written by two-photon polymerisation of a sol-gel resin using a femtosecond laser beam which was shaped using internal conical diffraction. The ring structure was written using a demagnified image of the ring-shaped beam which arises in conical diffraction of a narrow light beam. The pillar and wall structures were produced by imaging the Bessel beam formed by conical diffraction in combination with a converging lens.     

Colored conical emission in BBO crystal induced by intense femtosecond pulses

We present the experimental and theoretical study of colored conical emission (CCE) during intense femtosecond pulses propagating in a β-barium borate (BBO) crystal. CCE revealed special nonlinear dynamics of the strong coupling between the fundamental and second harmonic pulses due to the spatiotemporal modulation instability (MI). The underlying physics of CCE involve three three-photon processes, according to which the beam angular spectra was calculated on the phase-matching conditions and accords with the experimental observation.