Contrasting routes to the Kostka coefficients of λ [boxvr] n (ln)-permutational module expansions for 6n8: Applications to NMR spin clusters under SU(mn)×Ln spin symmetries

Aspects of the higher-n λ( _n) permutational modules associated with Young subgroups of various highly-branched high-n fold algebras, which are pertinent to identical spin NMR clusters, are presented for λ [boxvr] _n (or λ [boxvR] n), aRota p-tuple or number partition; the method of optimal choice for deriving the Λ_[λ′] Kostka coefficients, found in {[λ′]} sets derived from λ permutational module expansions, rests on the ordering of the λ-(shape) to the self-associated diagram(s) in the dominance hierarchy. Hence, physical insight into these cage-cluster NMR systems is developed both from these properties and from the inter-related induced symmetries of GL(n, ) and _n groups. From these associated combinatorial, mapping or scalar invariant aspects of SU(mn)× _n symmetry, one may define the [A]_n( _n) systems of [AX]_n NMR problems in a general semi-topological limit. This corresponds to a high-n _n limit in which the individual spin cluster exhibits a lack of any (intracluster) ‘magnetic equivalence’ properties.     

Identifying minimal and dominant solutions for Kummer recursions

We identify minimal and dominant solutions of three-term recurrence relations for the confluent hypergeometric functions and , where (not both equal to 0). The results are obtained by applying Perron's theorem, together with uniform asymptotic estimates derived by T. M. Dunster for Whittaker functions with large parameter values. The approximations are valid for complex values of , and , with .

     

Asymptotic expansions for Riesz fractional derivatives of Airy functions and applications

Riesz fractional derivatives of a function, (also called Riesz potentials), are defined as fractional powers of the Laplacian. Asymptotic expansions for large x are computed for the Riesz fractional derivatives of the Airy function of the first kind, Ai(x), and the Scorer function, Gi(x). Reduction formulas are provided that allow one to express Riesz potentials of products of Airy functions, and , via and . Here Bi(x) is the Airy function of the second type. Integral representations are presented for the function A₂(a,b;x)=Ai(x−a)Ai(x−b) with a,b∈R and its Hilbert transform. Combined with the above asymptotic expansions they can be used for computing asymptotics of the Hankel transform of . These results are used for obtaining the weak rotation approximation for the Ostrovsky equation (asymptotics of the fundamental solution of the linearized Cauchy problem as the rotation parameter tends to zero).     

Numerical evaluation of functions arising from transformations of formal series

An algorithm is given for the numerical evaluation of a class of functions of the confluent hypergeometric type. The method of computation is based on the well-known Miller algorithms and on asymptotic expansions.     

Cooperative Liouvillian mapping over carrier subspaces and induced of NMR spin clusters as a consequence of simple reducibility being applicable to tensorial operator bases,

The significance of simple-reducibility under groups being applicable to Liouville spin space {|k q v»} is considered in the context of induced symmetry and of mappings arising under cooperative symmetries. Some consequences of the realization of a further Heisenberg algebra, now under the {?_±, ?₀} superoperators, are considered in terms of mappings and Gel'fand pattern algebras over the distinct carrier space, . The cooperative aspects of induced symmetry are shown to be a consequence of fundamental Wigner superoperators and the transformational properties of an additional Yamanouchi permutation index being applicable over subspaces. The inherent structure of Liouville space under a more general symmetry is summarized from the viewpoint of lexical combinatorial treatments of the (k ₁?k _n ) -fields inherent in (Rota)-Cayley algebra.     

Preparing projected entangled pair states on a quantum computer

We present a quantum algorithm to prepare injective projected entangled pair states (PEPS) on a quantum computer, a class of open tensor networks representing quantum states. The run time of our algorithm scales polynomially with the inverse of the minimum condition number of the PEPS projectors and, essentially, with the inverse of the spectral gap of the PEPS's parent Hamiltonian.     

Analytical methods for an elliptic singular perturbation problem in a circle

We consider an elliptic perturbation problem in a circle by using the analytical solution that is given by a Fourier series with coefficients in terms of modified Bessel functions. By using saddle point methods we construct asymptotic approximations with respect to a small parameter. In particular we consider approximations that hold uniformly in the boundary layer, which is located along a certain part of the boundary of the domain.     

Asymptotics and Numerics of Zeros of Polynomials That Are Related to Daubechies Wavelets

We give asymptotic approximations of the zeros of certain high degree polynomials. The zeros can be used to compute the filter coefficients in the dilation equations which define the compactly supported orthogonal Daubechies wavelets. Computational schemes are presented to obtain the numerical values of the zeros within high precision.