Semiclassical quantization of Bohr orbits in the helium atom

We use the complex WKB-Maslov method to construct the semiclassical spectral series corresponding to the resonance Bohr orbits in the helium atom. The semiclassical energy levels represented as the Rydberg tetra series correspond to the doubly symmetrically excited states of helium-like atoms. This level series contains the Rydberg triple series reported by Richter and Wintgen in 1991, which corresponds to the Z²⁺e⁻e⁻ configuration of electrons observed by Eichmann and his collaborators in experiments on the laser excitation of the barium atom in 1992. The lower-level extrapolation of the formula obtained for the semiclassical spectrum gives the value of the ground state energy, which differs by 6% from the experimental value obtained by Bergeson and his collaborators in 1998. We also calculate the fine structure of the semiclassical spectrum due to the spin-orbit and spin-spin interactions of electrons. __________ Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 151, No. 2, pp. 261–286, May, 2007.     

Angle- and time-resolved mass spectrometric study on pulsed laser ablation of an La-Ca-Mn-O target

Pulsed laser ablation (PLA) of an La₂O₃-CaO-MnO₂ target at 532 nm has been investigated by angle- and time-resolved quadrupole mass spectrometry. The results show that different kinds of metal oxides as well as metal ions and atoms are produced during the ablation at high laser fluence. The measured TOF spectra are fitted by multicomponent Maxwell-Boltzmann distribution with a stream velocity, which gives the translational energy of 6.34 and 0.43 eV for Mn⁺ ions and Mn atoms, respectively. It implies that ablated ions are mainly formed via a nonthermal process, while the neutral atoms mainly via a thermal one. The angular distributions of Mn + ions and Mn atoms can be described by a cos ⁿ θ and a bicosine function a cosθ+ (1-a)cos ⁿ θ, respectively. Possible mechanisms of laser ablation of La-Ca-Mn-0 are discussed. Project supported by the National Natural Science Foundation of China (Grant No. 29683001).     

Molecular lines and continuum from W51A (I) —CO and NH3 spectra and 3 mm continuum

As for the 5′ × 4′(∼llpc × 9pc) region centered at W51 lRSl the observations of the 3.4 mm continuum, CO (J = 1-0) line and simultaneous NH₃(1,1), (2,2), (3,3), (4,4) inverse lines were made for studying the massive star formation region located in the main spiral arms of the Galaxy. In the directions of W51 IRS1, IRS2 and el/e2 in 3.4 mm continuum, analyses of the line profiles show that the absorption lines of ammonia, which arise from the gas in front of the HII region, are red-shifted with respect to the emission lines, which arise from the surrounding cloud. Furthermore, a radiation transfer and statistical equilibrium calculation of ammonia molecules show that the densities increase by 3–10 times from the eastern border to the center. These points hint that the collapse is happening in the molecular cloud core obscured in optical wavelengths. The effects of the radiation fields from radio, infrared and UCHII sources is non-negligible on the excitation of various molecules (e.g. NH₃) within the circle of radius 40″ centered at IRS1. The profiles of the COJ = 1–0 line in the circle change from double peaks ( ∼ 60, ∼ 68 km. s^-1) to triple peaks, i.e. the component ∼53 km·s⁻¹, which associates with UCHII, also appears in the spectra. There are indications that the circle of radius 40″ centered at IRSI is a region of massive star forming activity     

Optimal Partition Trees

We revisit one of the most fundamental classes of data structure problems in computational geometry: range searching. Matoušek (Discrete Comput. Geom. 10:157–182, 1993) gave a partition tree method for d-dimensional simplex range searching achieving O(n) space and O(n ^1−1/d ) query time. Although this method is generally believed to be optimal, it is complicated and requires O(n ^1+ε ) preprocessing time for any fixed ε>0. An earlier method by Matoušek (Discrete Comput. Geom. 8:315–334, 1992) requires O(nlogn) preprocessing time but O(n ^1−1/d log ^O(1) n) query time. We give a new method that achieves simultaneously O(nlogn) preprocessing time, O(n) space, and O(n ^1−1/d ) query time with high probability. Our method has several advantages:

Orthogonal graphs of characteristic 2 and their automorphisms

The (singular) orthogonal graph O(2ν + δ,q) over a field with q elements and of characteristic 2 (where ν 1, and δ = 0,1 or 2) is introduced. When ν = 1, O(2 · 1,q), O(2 · 1 + 1,q) and O(2 · 1 + 2,q) are complete graphs with 2, q + 1 and q2 + 1 vertices, respectively. When ν 2, O(2ν + δ,q) is strongly regular and its parameters are computed. O(2ν + 1,q) is isomorphic to the symplectic graph Sp(2ν,q). The chromatic number of O(2ν + δ,q) except when δ = 0 and ν is odd is computed and the group of graph automo...     

Metric Flips with Calabi Ansatz

We study the limiting behavior of the K?hler–Ricci flow on \mathbbP(O_\mathbbPⁿ ?O_\mathbbPⁿ(-1)^?(m+1)){{\mathbb{P}(\mathcal{O}_{\mathbb{P}^n} \oplus \mathcal{O}_{\mathbb{P}^n}(-1)^{\oplus(m+1)})}} for m, n ≥ 1, assuming the initial metric satisfies the Calabi symmetry. We show that the flow either shrinks to a point, collapses to \mathbbPⁿ{{\mathbb{P}^n}} or contracts a subvariety of codimension m + 1 in the Gromov–Hausdorff sense. We also show that the K?hler–Ricci flow resolves a certain type of cone singularities in the Gromov–Hausdorff sense.     

Wave atoms and time upscaling of wave equations

We present a new geometric strategy for the numerical solution of hyperbolic wave equations in smoothly varying, two-dimensional time-independent periodic media. The method consists in representing the time-dependent Green’s function in wave atoms, a tight frame of multiscale, directional wave packets obeying a precise parabolic balance between oscillations and support size, namely wavelength ~(diameter).² Wave atoms offer a uniquely structured representation of the Green’s function in the sense that

Smoothed analysis of termination of linear programming algorithms

 We perform a smoothed analysis of a termination phase for linear programming algorithms. By combining this analysis with the smoothed analysis of Renegar's condition number by Dunagan, Spielman and Teng (http://arxiv.org/abs/cs.DS/0302011) we show that the smoothed complexity of interior-point algorithms for linear programming is O(m ³ log(m/Σ)). In contrast, the best known bound on the worst-case complexity of linear programming is O(m ³ L), where L could be as large as m. We include an introduction to smoothed analysis and a tutorial on proof techniques that have been useful in smoothed analyses. Received: December 10, 2002 / Accepted: April 28, 2003 Published online: June 5, 2003 Key words. smoothed analysis – linear programming – interior-point algorithms – condition numbers Mathematics Subject Classification (1991): 90C05, 90C51, 68Q25     

Stokes and Navier–Stokes problems in a half-space: the existence and uniqueness of solutions a priori nonconvergent to a limit at infinity

The Cauchy problem and the initial boundary value problem in the half-space of the Stokes and Navier–Stokes equations are studied. The existence and uniqueness of classical solutions (u, π) (considered at least C ² × C ¹ smooth with respect to the space variable and C ¹ × C ⁰ smooth with respect to the time variable) without requiring convergence at infinity are proved. A priori the fields u and π are nondecreasing at infinity. In the case of the Stokes problem, the existence, for any t > 0, and the uniqueness of solutions with kinetic field and pressure field are established for some β ∈ (0, 1) and γ ∈ (0, 1 − β). In the case of Navier–Stokes equations, the existence (local in time) and the uniqueness of classical solutions to the Navier–Stokes equations are shown under the assumption that the initial data are only continuous and bounded, by proving that, for any t ∈ (0, T), the kinetic field u(x, t) is bounded and, for any γ ∈ (0, 1), the pressure field π(x, t) is O(1 + |x| ^γ ). Bibliography: 20 titles. To V. A. Solonnikov on his 75th birthday Published in Zapiski Nauchnykh Seminarov POMI, Vol. 362, 2008, pp. 176–240.     

Sufficient conditions for the subexponential property of the convolution of two distributions

Conditions on the distributions of two independent nonnegative random variablesX andY are given for the sumX+Y to have a subexponential distribution, i.e., (1−F ^(2*)(t))/(1−F(t)) → 2 ast → +∞, whereF(t)=P{X+Y≤t} andF ^(2*)(t) is the convolution ofF(t) with itself. Translated fromMatematicheskie Zametki, Vol. 58, No. 5, pp. 778–781, November, 1995.     

Quantum Fractals on <Emphasis Type="Italic">n</Emphasis>–Spheres. Clifford Algebra Approach

Using the Clifford algebra formalism we extend the quantum jumps algorithm of the Event Enhanced Quantum Theory (EEQT) to convex state figures other than those stemming from convex hulls of complex projective spaces that form the basis for the standard quantum theory. We study quantum jumps on n-dimensional spheres, jumps that are induced by symmetric configurations of non-commuting state monitoring detectors. The detectors cause quantum jumps via geometrically induced conformal maps (M?bius transformations) and realize iterated function systems (IFS) with fractal attractors located on n-dimensional spheres. We also extend the formalism to mixed states, represented by “density matrices” in the standard formalism, (the n-balls), but such an extension does not lead to new results, as there is a natural mechanism of purification of states. As a numerical illustration we study quantum fractals on the circle (one-dimensional sphere and pentagon), two–sphere (octahedron), and on three-dimensional sphere (hypercubetesseract, 24 cell, 600 cell, and 120 cell). The attractor, and the invariant measure on the attractor, are approximated by the powers of the Markov operator. In the appendices we calculate the Radon-Nikodym derivative of the SO(n + 1) invariant measure on Sⁿ under SO(1, n + 1) transformations and discuss the Hamilton’s “icossian calculus” as well as its application to quaternionic realization of the binary icosahedral group that is at the basis of the 600 cell and its dual, the 120 cell. As a by-product of this work we obtain several Clifford algebraic results, such as a characterization of positive elements in a Clifford algebra as generalized Lorentz “spin–boosts”, and their action as M?bius transformation on n-sphere, and a decomposition of any element of Spin⁺(1, n + 1) into a spin–boost and a spin–rotation, including the explicit formula for the pullback of the SO(n + 1) invariant Riemannian metric with respect to the associated M?bius transformation.     

Modified Lagrange–Galerkin methods of first and second order in time for convection–diffusion problems

We introduce modified Lagrange–Galerkin (MLG) methods of order one and two with respect to time to integrate convection–diffusion equations. As numerical tests show, the new methods are more efficient, but maintaining the same order of convergence, than the conventional Lagrange–Galerkin (LG) methods when they are used with either P ₁ or P ₂ finite elements. The error analysis reveals that: (1) when the problem is diffusion dominated the convergence of the modified LG methods is of the form O(h ^m+1 + h ² + Δt ^q ), q = 1 or 2 and m being the degree of the polynomials of the finite elements; (2) when the problem is convection dominated and the time step Δt is large enough the convergence is of the form O(\frach^m+1Dt+h²+Dt^q){O(\frac{h^{m+1}}{\Delta t}+h^{2}+\Delta t^{q})} ; (3) as in case (2) but with Δt small, then the order of convergence is now O(h ^m  + h ² + Δt ^q ); (4) when the problem is convection dominated the convergence is uniform with respect to the diffusion parameter ν (x, t), so that when ν → 0 and the forcing term is also equal to zero the error tends to that of the pure convection problem. Our error analysis shows that the conventional LG methods exhibit the same error behavior as the MLG methods but without the term h ². Numerical experiments support these theoretical results.     

EPR study of ZnS:Mn2+ nanocrystals and pyrex glasses

Pyrex glasses with different ZnS: Mn²⁺ contents were prepared by melting method. It has been found that Mn ion may occupy two sites: (Mn²⁺)_sub, and (Mn²⁺)_int from photoluminescene (PL) and photoluminescence excitation (PLE) spectra. The results were confirmed by the further electron panmagnetic resonance (EPR) experiments and the three types of states (Mn²⁺)_sub, (Mn²⁺)_int,and Mn clusters were identified. It was observed that theg-factor and the hyperfine structure (HFS) constant increase with the decreasing size of nanocrystallite. This may result from hybridization of sp³ electron states of ZnS and 3d⁵ electron states of Mn by the effects of quantum confinement and the surface states. Project supported by the National Natural Science Foundation of China and Laboratory of Excited State Processes, Chinese Academy of Sciences.     

Topological Model Categories Generated by Finite Complexes

 Our main result states that for each finite complex L the category TOP of topological spaces possesses a model category structure (in the sense of Quillen) whose weak equivalences are precisely maps which induce isomorphisms of all [L]-homotopy groups. The concept of [L]-homotopy has earlier been introduced by the first author and is based on Dranishnikov’s notion of extension dimension. As a corollary we obtain an algebraic characterization of [L]-homotopy equivalences between [L]-complexes. This result extends two classical theorems of J. H. C. Whitehead. One of them – describing homotopy equivalences between CW-complexes as maps inducing isomorphisms of all homotopy groups – is obtained by letting L = {point}. The other – describing n-homotopy equivalences between at most (n+1)-dimensional CW-complexes as maps inducing isomorphisms of k-dimensional homotopy groups with k ⩽ n – by letting L = S ⁿ⁺¹ , n ⩾ 0.     

Analysis of a new class of forward semi-Lagrangian schemes for the 1D Vlasov Poisson equations

The Vlasov equation is a kinetic model describing the evolution of a plasma which is a globally neutral gas of charged particles. It is self-consistently coupled with Poisson’s equation, which rules the evolution of the electric field. In this paper, we introduce a new class of forward semi-Lagrangian schemes for the Vlasov–Poisson system based on a Cauchy Kovalevsky (CK) procedure for the numerical solution of the characteristic curves. Exact conservation properties of the first moments of the distribution function for the schemes are derived and a convergence study is performed that applies as well for the CK scheme as for a more classical Verlet scheme. A L ¹ convergence of the schemes will be proved. Error estimates [in O(Dt²+h² + \frach²Dt){O\left(\Delta{t}^2+h^2 + \frac{h^2}{\Delta{t}}\right)} for Verlet] are obtained, where Δt and h = max(Δx, Δv) are the discretization parameters.     

The abc-conjecture for Algebraic Numbers

The abe-conjecture for the ring of integers states that, for every ε 〉 0 and every triple of relatively prime nonzero integers （a, b, c） satisfying a ＋ b = c, we have max（｜a｜, ｜b｜, ｜c｜） 〈 rad（abc）^1＋ε with a finite number of exceptions. Here the radical rad（m） is the product of all distinct prime factors of m. In the present paper we propose an abe-conjecture for the field of all algebraic numbers. It is based on the definition of the radical （in Section 1） and of the height （in Section 2） of an algebraic number. From this abc-conjecture we deduce some versions of Fermat＇s last theorem for the field of all algebraic numbers, and we discuss from this point of view known results on solutions of Fermat＇s equation in fields of small degrees over Q.     

FFTs on the Rotation Group

We discuss an implementation of an efficient algorithm for the numerical computation of Fourier transforms of bandlimited functions defined on the rotation group SO(3). The implementation is freely available on the web. The algorithm described herein uses O(B ⁴) operations to compute the Fourier coefficients of a function whose Fourier expansion uses only (the O(B ³)) spherical harmonics of degree at most B. This compares very favorably with the direct O(B ⁶) algorithm derived from a basic quadrature rule on O(B ³) sample points. The efficient Fourier transform also makes possible the efficient calculation of convolution over SO(3) which has been used as the analytic engine for some new approaches to searching 3D databases (Funkhouser et al., ACM Trans. Graph. 83–105, [2003]; Kazhdan et al., Eurographics Symposium in Geometry Processing, pp. 167–175, [2003]). Our implementation is based on the “Separation of Variables” technique (see, e.g., Maslen and Rockmore, Proceedings of the DIMACS Workshop on Groups and Computation, pp. 183–237, [1997]). In conjunction with techniques developed for the efficient computation of orthogonal polynomial expansions (Driscoll et al., SIAM J. Comput. 26(4):1066–1099, [1997]), our fast SO(3) algorithm can be improved to give an algorithm of complexity O(B ³log ² B), but at a cost in numerical reliability. Numerical and empirical results are presented establishing the empirical stability of the basic algorithm. Examples of applications are presented as well. First author was supported by NSF ITR award; second author was supported by NSF Grant 0219717 and the Santa Fe Institute.     

Quantum integrability and quantum chaos in the micromaser

The time-dependent quantum Hamiltonians

Local and parallel algorithms for fourth order problems discretized by the Morley–Wang–Xu element method

This paper systematically studies numerical solution of fourth order problems in any dimensions by use of the Morley–Wang–Xu (MWX) element discretization combined with two-grid methods (Xu and Zhou (Math Comp 69:881–909, 1999)). Since the coarse and fine finite element spaces are nonnested, two intergrid transfer operators are first constructed in any dimensions technically, based on which two classes of local and parallel algorithms are then devised for solving such problems. Following some ideas in (Xu and Zhou (Math Comp 69:881–909, 1999)), the intrinsic derivation of error analysis for nonconforming finite element methods of fourth order problems (Huang et al. (Appl Numer Math 37:519–533, 2001); Huang et al. (Sci China Ser A 49:109–120, 2006)), and the error estimates for the intergrid transfer operators, we prove that the discrete energy errors of the two classes of methods are of the sizes O(h + H ²) and O(h + H ²(H/h)^(d−1)/2), respectively. Here, H and h denote respectively the mesh sizes of the coarse and fine finite element triangulations, and d indicates the space dimension of the solution region. Numerical results are performed to support the theory obtained and to compare the numerical performance of several local and parallel algorithms using different intergrid transfer operators.     

Gauge-Invariant Characterization of Yang–Mills–Higgs Equations

Let C → M be the bundle of connections of a principal G-bundle P → M over a pseudo-Riemannian manifold (M,g) of signature (n⁺, n⁻) and let E → M be the associated bundle with P under a linear representation of G on a finite-dimensional vector space. For an arbitrary Lie group G, the O(n⁺, n⁻) × G-invariant quadratic Lagrangians on J¹(C × _M E) are characterized. In particular, for a simple Lie group the Yang–Mills and Yang–Mills–Higgs Lagrangians are characterized, up to an scalar factor, to be the only O(n⁺, n⁻) × G-invariant quadratic Lagrangians. These results are also analyzed on several examples of interest in gauge theory. Submitted: May 19, 2005; Accepted: April 25, 2006