Infinite order symmetries for two-dimensional separable Schrödinger equations
W. MILLER †, E. KALNINS ‡ and G. POGOSYAN § † School of Mathematics, University of Minnesota, Minneapolis, Minnesota, 55455, U.S.A.  miller@ima.umn.edu ‡ Department of Mathematics, University of Waikato, Hamilton, New Zealand E-mail: e.kalnins@waikato.ac.nz § Laboratory of Theoretical Physics, Joint Institute for Nuclear Research, 14980 Dubna, Moscow Region, Russia E-mail: pogosyan@thsun1.jinr.dubna.su

title: Lie theory and separation of variables. 11. The EPD equation
abstract: We show that the Euler–Poisson–Darboux equation {∂tt−∂rr−[(2m+1)/r]∂r}Θ=0 separates in exactly nine coordinate systems corresponding to nine orbits of symmetric second‐order operators in the enveloping algebra of SL(2,R), the symmetry group of this equation. We employ techniques developed in earlier papers from this series and use the representation theory of SL(2,R) to derive special function identities relating the separated solutions. We also show that the complex EPD equation separates in exactly five coordinate systems corresponding to five orbits of symmetric second‐order operators in the enveloping algebra of SL(2,C).

title: CLEBSCH--GORDAN COEFFICIENTS AND SPECIAL FUNCTION IDENTITIES. II. THE ROTATION AND LORENTZ GROUPS IN 3-SPACE.
abstract: It is shown that the construction of concrete models of Clebsch‐Gordan decompositions for tensor products of irreducible group representations leads to a wide variety of special function identities. In this paper the representation theory of the rotation and Lorentz groups in 3‐space is used to give elegant derivations of identities involving Languerre, Gegenbauer, hypergeometric, and generalized hypergeometric functions. Some of these identities may be new in this general form.

title: Lie theory and separation of variables. 7. The harmonic oscillator in elliptic coordinates and Ince polynomials
abstract: As a continuation of Paper 6 we study the separable basis eigenfunctions and their relationships for the harmonic oscillator Hamiltonian in two space variables with special emphasis on products of Ince polynomials, the eigenfunctions obtained when one separates variables in elliptic coordinates. The overlaps connecting this basis to the polar and Cartesian coordinate bases are obtained by computing in a simpler Bargmann Hilbert space model of the problem. We also show that Ince polynomials are intimately connected with the representation theory of SU (2), the group responsible for the eigenvalue degeneracy of the oscillator Hamiltonian.

title: Separation of variables in Einstein spaces. I. Two ignorable and one null coordinate
abstract: Computes and classifies all Einstein spaces and their corresponding coordinate systems such that the Hamilton-Jacobi equation admits a separation of variables with two ignorable and one essential null coordinates. The authors find that all such spaces correspond to plane-fronted gravitational waves of Petrov type N. They also give necessary conditions which a general space-time must have in order to admit coordinates of the type studied. Furthermore, the integrability conditions for Helmholtz (Schrodinger) separability are completely solved.

title: Complete sets of commuting operators and O (3) scalars in the enveloping algebra of SU (3)
abstract: We consider the ``missing label'' problem for basis vectors of SU(3) representations in a basis corresponding to the group reduction SU(3)⊃O(3)⊃O(2). We prove that only two independent O (3) scalars exist in the enveloping algebra of S U (3), in addition to the obvious ones, namely the angular momentum L2 and the two SU(3) Casimir operators C(2) and C(3). Any one of these two operators (of third and fourth order in the generators) can be added to C(2), C(3), L2, and L3 to form a complete set of commuting operators. The eigenvalues of the third and fourth order scalars X(3) and X(4) are calculated analytically or numerically for many cases of physical interest. The methods developed in this article can be used to resolve a missing label problem for any semisimple group G, when reduced to any semisimple subgroup H.

title: Simultaneous observations of extensive air showers and underground muons at Soudan 2
abstract: We have built a small proportional tube air shower array on the surface above the Soudan 2 proton decay detector in Soudan, Minnesota, USA. This array, in coincidence with the underground detector, allows simultaneous surface and underground observations of high energy cosmic ray showers. These observations, still in their initial stages, may eventually be able to provide information about the composition of cosmic ray primaries in the knee'' region of the energy spectrum. 4 refs.

title: First direct observation of muon antineutrino disappearance.
abstract: This Letter reports the first direct observation of muon antineutrino disappearance. The MINOS experiment has taken data with an accelerator beam optimized for ν(μ) production, accumulating an exposure of 1.71 × 10²⁰ protons on target. In the Far Detector, 97 charged current ν(μ) events are observed. The no-oscillation hypothesis predicts 156 events and is excluded at 6.3σ. The best fit to oscillation yields |Δm²| = [3.36(-0.40)(+0.46)(stat) ± 0.06(syst)] × 10⁻³ eV², sin²(2θ) = 0.86(-0.12)(+0.11)(stat) ± 0.01(syst). The MINOS ν(μ) and ν(μ) measurements are consistent at the 2.0% confidence level, assuming identical underlying oscillation parameters.

title: RESONANCES IN THE SCATTERING OF ELECTRONS FROM ATOMS.

title: ATLAS pixel detector electronics and sensors
abstract: The silicon pixel tracking system for the ATLAS experiment at the Large Hadron Collider is described and the performance requirements are summarized. Detailed descriptions of the pixel detector electronics and the silicon sensors are given. The design, fabrication, assembly and performance of the pixel detector modules are presented. Data obtained from test beams as well as studies using cosmic rays are also discussed.

title: Search for flavor-changing non-standard neutrino interactions by MINOS
abstract: We report new constraints on flavor-changing non-standard neutrino interactions from the MINOS experiment, in which neutrino versus antineutrino interactions can be distinguished on an event-by-event basis. We analyzed a combined set of beam neutrino and antineutrino data from the well-understood NuMI beam, and found no evidence for deviations from standard neutrino mixing. The observed energy spectra constrain the non-standard neutrino interactions parameter to the range −0.20