Aldo Cossu’s Work in Finite Geometry

A survey of the contributions of Aldo Cossu in finite geometry is given. Dedicated to the memory of Professor Aldo Cossu     

自由曲面光学虚拟制造与检测系统的探讨

自由曲面光学产品设计、制造与检测的工艺流程，通常采取试凑法逐次逼近。由于加工 检测 再加工，循环往复，既费时，成本又高，产生了瓶颈问题。为了解决此弊端，本文运用虚拟制造技术，提出光学虚拟制造的基本构想，即虚拟制造系统结构模型，给出光学系统虚拟原型的构成和光学系统成像质量虚拟检测系统的构成，讨论光学成像质量的仿真检测以及敏度分析方法。研究结果表明：运用虚拟制造与检测技术，可缩短研发周期，降低成本，优化工艺并提高产品质量。     

Polarizability of the fine-structure components of low excited states of the F, Cl, and Br atoms

The dynamic scalar, tensor, and pseudovector polarizabilities of the low excited states (ns)⁴ P _J of the F, Cl, and Br atoms (n = 3, 4, and 5, respectively) are calculated for all the levels of the fine-structure multiplets with J = 5/2, 3/2, and 1/2 with the use of the quantum detect Green’s function, proposed earlier.     

Mathematical modeling of random coupling between polarization modes in single-mode optical fibers: XIII. Conditions of applicability of the ergodic hypothesis for fiber ring interferometers

Some additional conditions of applicability of the ergodic hypothesis to fiber ring interferometers (FRIs) with a loop consisting of a single-mode optical fiber (SMOF) with random inhomogeneities are considered. It is shown by mathematical modeling that the change in the phase difference of counterpropagating waves at the FRI output with the SMOF temperature is not a stationary random process. However, in a fairly narrow temperature range, this dependence can be assumed to be locally stationary. The conditions determining this temperature range are formulated. It is shown for a fairly large ensemble of independent realizations of random inhomogeneities in an SMOF that, even when all conditions of ergodicity are satisfied with a large margin, there will always be at least one realization violating strict ergodicity. Thus, only conditional (approximate) ergodicity occurs in this case. Nevertheless, in calculation of the FRI zero drift in this situation, averaging over an ensemble of independent realizations of random inhomogeneities in the SMOF of an FRI loop can be performed with sufficient accuracy. As a result, calculations are simplified significantly. In the general case, when at least one of the conditions of ergodicity is not satisfied, averaging over temperature for each realization with subsequent averaging over the entire ensemble should be performed. It is shown also that, within this problem, we can speak only about quasi-ergodicity or emulation of ergodicity, since a change in the temperature of the SMOF of an FRI loop and successive enumeration of independent realizations of random inhomogeneities in the SMOF loop are radically different random processes. The parameters characterizing quasiperiodic temperature changes in the phase difference of counterpropagating waves at the FRI output are refined.     

Cyclotron resonance for electrons over helium in a resonator

The cyclotron resonance (CR) problem for electrons over a helium film occupying the lower part of a resonator is solved. This problem is shown to represent an example of the well-known problem on the behavior of a system of coupled oscillators. For such oscillators, the coupling constant is determined as a function of the problem parameters with its minimal value in zero magnetic field and its maximal value at resonance conditions, when the cyclotron frequency coincides with one of the resonator modes. The details of the CR absorption of microwave energy by the coupled system formed by 2D electrons and a resonator are calculated. The results are discussed in application to the known CR experiments with electrons over helium.     

Absorption and emission dynamics in concentrated optical ensembles under laser excitation

A new theoretical model describing the emission and absorption dynamics in an ensemble of molecules under intense coherent pulsed pumping is developed on the basis of the concepts of cooperative light-induced luminescence (CLIL). The CLIL development is described within the framework of formalism of the system density matrix in the space of photon wave functions. It is shown that the fast growth of CLIL relates to the development of coherent states of the quantum field in the area of efficient cooperative interactions of molecules (coherence volume). A system of equations for the calculation of CLIL energy, population of excited states, and optical absorption of the system in dependence on the laser pump energy density is solved. The theoretical results obtained are in good agreement with the experimental data.     

An attempt on the calculation of relativistic potential energy curves: Noble gas difluorides

Total energy SCF calculations were performed for noble gas difluorides in a relativistic procedure and compared with analogous non-relativistic calculations. The discrete variational method with numerical basis functions was used. Rather smooth potential energy curves could be obtained. The theoretical Kr-F and Xe-F bond distances were calculated to be 3.5 a.u. and 3.6 a.u. which should be compared with the experimental values of 3.54 a.u. and 3.7 a.u. Although the dissociation energies are off by a factor of about five it was found that ArF₂ may be a stable molecule. Theoretical ionization energies for the outer levels reproduce the experimental values for KrF₂ and XeF₂ to within 2 eV.     

Pure adaptive search in monte carlo optimization

Pure adaptive search constructs a sequence of points uniformly distributed within a corresponding sequence of nested regions of the feasible space. At any stage, the next point in the sequence is chosen uniformly distributed over the region of feasible space containing all points that are equal or superior in value to the previous points in the sequence. We show that for convex programs the number of iterations required to achieve a given accuracy of solution increases at most linearly in the dimension of the problem. This compares to exponential growth in iterations required for pure random search.     

Semiclassical Orthogonal Polynomials, Matrix Models and Isomonodromic Tau Functions

The differential systems satisfied by orthogonal polynomials with arbitrary semiclassical measures supported on contours in the complex plane are derived, as well as the compatible systems of deformation equations obtained from varying such measures. These are shown to preserve the generalized monodromy of the associated rank-2 rational covariant derivative operators. The corresponding matrix models, consisting of unitarily diagonalizable matrices with spectra supported on these contours are analyzed, and it is shown that all coefficients of the associated spectral curves are given by logarithmic derivatives of the partition function or, more generally, the gap probabilities. The associated isomonodromic tau functions are shown to coincide, within an explicitly computed factor, with these partition functions. Research supported in part by the Natural Sciences and Engineering Research Council of Canada, the Fonds FCAR du Québec and EC ITH Network HPRN-CT-1999-000161.     

Cusp catastrophe in slowly varying equilibriums

Solutions of the equations v _x+v ³?tv+x=0 and v _xx=v ³? tv+x, which describe the nucleation of domain walls occurring in the neighborhood of cusps of slowly varying equilibriums, are analyzed. Examples related to the diffusion in smoothly inhomogeneous media are considered.