[en] It is shown that vibrational displacements satisfying the Eckart−Sayvetz conditions can be constructed by projection of unconstrained displacements. This result has a number of interesting direct and indirect ramifications: (i) The normal coordinates corresponding to an electronic state or an isotopologue of a molecule are transformed to those of another state or isotopologue by a linear and, in general, non-orthogonal transformation. (ii) Novel interpretation of axis switching. (iii) One may enhance the separation of rotational-large-amplitude internal motions and the vibrational motions beyond that offered by the standard use of the Eckart−Sayvetz conditions. (iv) The rotational-vibrational Hamiltonian given in terms of curvilinear internal coordinates may be derived with elementary mathematical tools while taking into account the Eckart conditions with or without enhancement

[en] A multivariable generalization is presented for all the discrete families of the Askey tableau. This significantly extends the multivariable Hahn polynomials introduced by Karlin and McGregor. The latter are recovered as a limit case from a family of multivariable Racah polynomials

[en] We study the isometry groups of semi-orthogonal forms (that is, forms whose Gram matrix in some basis is upper triangular with ones on the diagonal) on a Z-module of rank 3. Such forms have a discrete parameter: the height (the trace of the dualizing operator + 3). We prove that the isometry group is either Z or Z₂×Z, list all the cases when it is a direct product and describe the generator of order 2 in that case. We also describe a generator of infinite order for many particular values of the height.

[en] This paper begins with an exposition of a systematic technique for generating orthonormal polynomials in two independent variables by application of the Gram-Schmidt orthogonalization procedure of linear algebra. It is then demonstrated how a linear least squares approximation for experimental data or an arbitrary function can be generated from these polynomials. The least squares coefficients are computed without recourse to matrix arithmetic, which ensures both numerical stability and simplicity of implementation as a self contained numerical algorithm. The Gram-Schmidt procedure is then utilised to generate a complete set of orthogonal polynomials of fourth degree. A theory for the transformation of the polynomial representation from an arbitrary basis into the familiar sum of products form is presented, together with a specific implementation for fourth degree polynomials. Finally, the computational integrity of this algorithm is verified by reconstructing arbitrary fourth degree polynomials from their values at randomly chosen points in their domain. 13 refs., 1 tab

[en] Transformation of a C-point singularity into its orthogonal state has been introduced recently. This non-trivial process of transformation naturally raises a question about its significance. In this paper we show that orthogonal C-point polarization singularities can be used to construct basis sets of polarization optics. The generic C-points namely lemons and stars are used to substantiate the idea. Here, we show homogeneous polarization distributions as superpositions of two orthogonal C-points. The Wronskian of the two orthogonal basis states is non-zero which indicates that the basis states are linearly independent. This novel idea where homogeneous polarization distributions are expressed as superpositions of singular polarization distributions may be useful in diverse fields beyond polarization optics. This exercise is shown to bring out hitherto unknown interesting structures like interconnections between polarization singularity, phase singularity and stereographic projection, in polarization optics. (paper)

[en] Via the solutions of systems of algebraic equations of Bethe Ansatz type, we arrive at bounds for the zeros of orthogonal (basic) hypergeometric polynomials belonging to the Askey–Wilson, Wilson and continuous Hahn families.

[en] In this Reply to the preceding Comment by Hall and Rao [Phys. Rev. A 83, 036101 (2011)], we motivate terminology of our original paper and point out that further research is needed in order to (dis)prove the claimed link between every orthogonal Latin square of order being a power of a prime and a mutually unbiased basis.

[en] The 2-cohomology group is determined for the finite simple orthogonal group Ω^-(4,q), where q is odd, with coefficients in the natural module. For q≠9 this group is trivial, and for q=9 it is isomorphic to Z₃⁴. Thus Kuesefoglu's result is corrected. Bibliography: 5 titles.

[en] We prove that for |x| ⩽ |t| < 1, −1 < q ⩽ 1 and n ⩾ 0: Σ_≥(tⁱ)/(q)_ih_n+i( x|q) =h_n(x|t,q) Σ_i≥0(tⁱ)/(q)_ih_i(x|q), where h_n(x|q) and h_n(x|t, q) are respectively the so-called q-Hermite and the big q-Hermite polynomials, and (q)_n denotes the so-called q-Pochhammer symbol. We prove similar equalities involving big q-Hermite and Al-Salam–Chihara polynomials, and Al-Salam–Chihara and the so-called continuous dual q-Hahn polynomials. Moreover, we are able to relate in this way some other ‘ordinary’ orthogonal polynomials such as, e.g., Hermite, Chebyshev or Laguerre. These equalities give a new interpretation of the polynomials involved and moreover can give rise to a simple method of generating more and more general (i.e. involving more and more parameters) families of orthogonal polynomials. We pose some conjectures concerning Askey–Wilson polynomials and their possible generalizations. We prove that these conjectures are true for the cases q = 1 (classical case) and q = 0 (free case), thus paving the way to generalization of Askey–Wilson polynomials at least in these two cases. (paper)

[en] In a recent article Paterek, Dakic, and Brukner [Phys. Rev. A 79, 012109 (2009)] show an algorithm for generating mutually unbiased bases from sets of orthogonal Latin squares. They claim that this algorithm works for every set of orthogonal Latin squares. We show that the algorithm only works for particular sets of orthogonal Latin squares. Furthermore, the algorithm is a more readable version of work previously published [Phys. Rev. A 70, 062101 (2004)].