The Asymptotic and Numerical Inversion of the Marcum Q‐Function

The generalized Marcum functions appear in problems of technical and scientific areas such as, for example, radar detection and communications. In mathematical statistics and probability theory these functions are called the noncentral gamma or the noncentral chi‐squared cumulative distribution functions. In this paper, we describe a new asymptotic method for inverting the generalized Marcum Q‐function and for the complementary Marcum P‐function. Also, we show how monotonicity and convexity properties of these functions can be used to find initial values for reliable Newton or secant methods to invert the function. We present details of numerical computations that show the reliability of the asymptotic approximations.     

First Order Approximation of an Elliptic 3D Singular Perturbation Problem

A three-dimensional elliptic singular perturbation problem with discontinuous boundary values is considered. The solution of the problem is written in terms of a double integral. A saddle point analysis is used to obtain a first approximation, which is expressed in terms of a function that can be viewed as a generalization of the complementary error function.     

Symbolic evaluation of coefficients in Airy-type asymptotic expansions

Computer algebra algorithms are developed for evaluating the coefficients in Airy-type asymptotic expansions that are obtained from integrals with a large parameter. The coefficients are defined from recursive schemes obtained from integration by parts. An application is given for the Weber parabolic cylinder function.     

New asymptotic representations of the noncentral t-distribution

New asymptotic approximations of the noncentral t distribution are given a generalization of the Student's t distribution. Using new integral representations, we give new asymptotic expansions not only for large values of the noncentrality parameter but also for large values of the degrees of freedom parameter. In some cases, we accept more than one large parameter. These results are not only in terms of elementary functions, but also in terms of the complementary error function and the incomplete gamma function. A number of numerical tests demonstrate the performance of the asymptotic approximations.     

On the zeros of the Scorer functions

Asymptotic approximations are developed for zeros of the solutions Gi(z) and Hi(z) of the inhomogeneous Airy differential equation . The solutions are also called Scorer functions. Tables are given with numerical values of the zeros.     

Neutron scattering from liquid crystal mesophases: Form of the elastic incoherent structure factor for “monocrystal” aligned nematogen with specific Q to n orientation

The elastic incoherent structure function (ElSF), A_o(Q), is derived for specific experimental orientations of the momentum transfer vector, Q, with respect to the nematic director, n, of an aligned “monocrystal” mesophase. The particular uniaxial model first considered is that in which the initial probability component of the orientational correlation function departs from uniformity to become a peaked function. The analytical EISF for Q||n is derived on the basis of the microscopic scattering symmetry and an effectively static principal axis. Brief explicit consideration is given to the EISF for the more general “dynamic principal axis” model with specific Q to an experimental configuration. An alternative closed form is given for S_k (δ′), the dynamical equivalent of an order parameter.     

Graphene with geometrically induced vorticity

At half filling, the electronic structure of graphene can be modeled by a pair of free two-dimensional Dirac fermions. We explicitly demonstrate that in the presence of a geometrically induced gauge field an everywhere-real Kekulé modulation of the hopping matrix elements can correspond to a nonreal Higgs field with nontrivial vorticity. This provides a natural setting for fractionally charged vortices with localized zero modes. For fullerenelike molecules we employ the index theorem to demonstrate the existence of six low-lying states that do not depend strongly on the Kekulé-induced mass gap.     

The Astounding Chemistry of a 2‐Amino‐1,2‐dihydroisoquinoline Derivative

The cycloadducts of isoquinolinium N‐phenyl imide 2 with C=C bonds are derivatives of 2‐amino‐1,2‐dihydroisoquinoline. Their N^β‐vinylphenylhydrazine system is amenable to an acid‐catalyzed [3,3]‐sigmatropic shift; the formation of pentacyclic aminals is exemplified by 6 → 8 . The dimethyl maleate adduct 11 , C₂₁H₂₀N₂O₄, is exceptional by being converted on treatment with acid to bright‐yellow crystals, C₂₄H₂₂N₂O₆ (additional C₃H₂O₂). X‐Ray crystal‐structure analysis and NMR spectra reveal structure 13 , and mechanistic studies indicated an initial β‐elimination at the N−N bond of 11 to yield 18 ; this step is followed by a retro‐Mannich‐type cleavage that gives methyl isoquinoline‐1‐acetate ( 14 ) and methyl 2‐(phenylimino)acetate ( 15 ), according to the sequence C₂₁H₂₀N₂O₄ ( 11 )→ 18 →C₁₂H₁₁NO₂ ( 14 )+C₉H₉NO₂ ( 15 ). In the second act of the drama, electrophilic attack by 15 ‐H⁺ on the ene‐hydrazine group of a second molecule of 11 furnishes 13 by a polystep intramolecular redox reaction. All rate constants must be fine‐tuned in this reaction cascade to give 13 in yields of up to 78% with an overall stoichiometry: 2 C₂₁H₂₀N₂O₄ ( 11 )→C₂₄H₂₂N₂O₆ ( 13 )+C₁₂H₁₁NO₂ ( 14 )+aniline. Interception and model experiments confirmed the above pathway. A by‐product, C₃₃H₃₁N₃O₆ ( 62 ), arises from an acid‐catalyzed dimerization of 11 and subsequent elimination of 15 .