CLEBSCH--GORDAN COEFFICIENTS AND SPECIAL FUNCTION IDENTITIES. II. THE ROTATION AND LORENTZ GROUPS IN 3-SPACE.
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: Superintegrability on the two-dimensional hyperboloid. II
abstract: In this work we examine the basis functions for classical and quantum mechanical systems on the two-dimensional hyperboloid that admit separation of variables in at least two coordinate systems. We present all of these cases from a unified point of view. In particular, all of the special functions that arise via variable separation have their essential features expressed in terms of their zeros. The principal new results are the details of the polynomial bases for each of the nonsubgroup bases, not just the subgroup spherical coordinate cases, and the details of the structure of the quadratic symmetry algebras.

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: 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: Search for Lorentz invariance and CPT violation with the MINOS far detector.
abstract: We searched for a sidereal modulation in the MINOS far detector neutrino rate. Such a signal would be a consequence of Lorentz and CPT violation as described by the standard-model extension framework. It also would be the first detection of a perturbative effect to conventional neutrino mass oscillations. We found no evidence for this sidereal signature, and the upper limits placed on the magnitudes of the Lorentz and CPT violating coefficients describing the theory are an improvement by factors of 20-510 over the current best limits found by using the MINOS near detector.

title: First observations of separated atmospheric nu(mu) and anti-nu(mu) events in the MINOS detector.
abstract: The complete 5.4 kton MINOS far detector has been taking data since the beginning of August 2003 at a depth of 2070 meters water-equivalent in the Soudan mine, Minnesota. This paper presents the first MINOS observations of muon neutrino and muon anti-neutrino charged-current atmospheric neutrino interactions based on an exposure of 418 days. The ratio of upward to downward-going events in the data is compared to the Monte Carlo expectation in the absence of neutrino oscillations giving: R_data(up/down)/R_MC(up/down) = 0.62^{+0.19}_{-0.14} (stat.) +- 0.02 (sys.). An extended maximum likelihood analysis of the observed L/E distributions excludes the null hypothesis of no neutrino oscillations at the 98 % confidence level. Using the curvature of the observed muons in the 1.3 T MINOS magnetic field muon neutrino and muon anti-neutrino interactions are separated. The ratio of muon neutrino to muon anti-neutrino events in the data is compared to the Monte Carlo expectation assuming neutrinos and anti-neutrinos oscillate in same manner giving: R_data(numubar/numu)/R_MC(numubar/numu) = 0.96^{+0.38}_{-0.27} (stat.) +- 0.15 (sys.), where the errors are the statistical and systematic uncertainties. Although the statistics are limited, this is the first direct observation of atmospheric neutrino interactions separately for muon neutrinos and muon anti-neutrinos.

title: Search for Lorentz Invariance and CPT Violation with Muon Antineutrinos in the MINOS Near Detector
abstract: We have searched for sidereal variations in the rate of antineutrino interactions in the MINOS Near Detector. Using antineutrinos produced by the NuMI beam, we find no statistically significant sidereal modulation in the rate. When this result is placed in the context of the Standard Model Extension theory we are able to place upper limits on the coefficients defining the theory. These limits are used in combination with the results from an earlier analysis of MINOS neutrino data to further constrain the coefficients.

title: Measurement of neutrino and antineutrino oscillations using beam and atmospheric data in MINOS.
abstract: We report measurements of oscillation parameters from ν(μ) and ν(μ) disappearance using beam and atmospheric data from MINOS. The data comprise exposures of 10.71×10(20) protons on target in the ν(μ)-dominated beam, 3.36×10(20) protons on target in the ν(μ)-enhanced beam, and 37.88 kton yr of atmospheric neutrinos. Assuming identical ν and ν oscillation parameters, we measure |Δm2| = (2.41(-0.10)(+0.09))×10(-3)  eV2 and sin2(2θ) = 0.950(-0.036)(+0.035). Allowing independent ν and ν oscillations, we measure antineutrino parameters of |Δm2| = (2.50(-0.25)(+0.23))×10(-3)  eV2 and sin2(2θ) = 0.97(-0.08)(+0.03), with minimal change to the neutrino parameters.

title: Testing Lorentz invariance and CPT conservation with NuMI neutrinos in the MINOS near detector.
abstract: A search for a sidereal modulation in the MINOS near detector neutrino data was performed. If present, this signature could be a consequence of Lorentz and CPT violation as predicted by the effective field theory called the standard-model extension. No evidence for a sidereal signal in the data set was found, implying that there is no significant change in neutrino propagation that depends on the direction of the neutrino beam in a sun-centered inertial frame. Upper limits on the magnitudes of the Lorentz and CPT violating terms in the standard-model extension lie between 10(-4) and 10(-2) of the maximum expected, assuming a suppression of these signatures by a factor of 10(-17).